diff --git a/src/cli/runtime/websocket-transport.test.ts b/src/cli/runtime/websocket-transport.test.ts
index 4c178b71fd8..ed4330c8f1a 100644
--- a/src/cli/runtime/websocket-transport.test.ts
+++ b/src/cli/runtime/websocket-transport.test.ts
@@ -18,6 +18,7 @@ import { launchOrcaApp } from './launch'
 import { addEnvironmentFromPairingCode } from './environments'
 import { RuntimeClientError } from './types'
 import {
+  AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
   AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY,
   AGENT_SESSION_BOUNDARY_RUNTIME_CAPABILITY,
   AUTOMATION_OWNER_FENCING_RUNTIME_CAPABILITY,
@@ -72,6 +73,7 @@ describe('CLI remote WebSocket transport', () => {
       expect.objectContaining({
         clientCapabilities: [
           AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY,
+          AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
           SESSION_TAB_CLOSE_INTENT_RUNTIME_CAPABILITY,
           SESSION_TABS_AUTHORITATIVE_INVENTORY_RUNTIME_CAPABILITY,
           AGENT_SESSION_BOUNDARY_RUNTIME_CAPABILITY,
diff --git a/src/main/ipc/runtime.ts b/src/main/ipc/runtime.ts
index 6237b8d040d..81de25d66c0 100644
--- a/src/main/ipc/runtime.ts
+++ b/src/main/ipc/runtime.ts
@@ -11,6 +11,7 @@ import type { RuntimeRpcResponse } from '../../shared/runtime-rpc-envelope'
 import type { ClientHostedBrowserRowsEvent } from '../../shared/client-hosted-browser-rows'
 import { TERMINAL_FIT_RESTORE_DEADLINE_MS } from '../../shared/terminal-fit-restore-deadline'
 import {
+  AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
   AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY,
   CLAUDE_STRUCTURED_AGENT_SESSION_RUNTIME_CAPABILITY,
   STRUCTURED_AGENT_SESSION_RUNTIME_CAPABILITY
@@ -82,6 +83,7 @@ export function registerRuntimeHandlers(runtime: OrcaRuntimeService): void {
           connectionId: desktopSenders.connectionIdFor(event.sender),
           clientCapabilities: [
             AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY,
+            AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
             STRUCTURED_AGENT_SESSION_RUNTIME_CAPABILITY,
             CLAUDE_STRUCTURED_AGENT_SESSION_RUNTIME_CAPABILITY
           ]
@@ -131,6 +133,7 @@ export function registerRuntimeHandlers(runtime: OrcaRuntimeService): void {
             connectionId,
             clientCapabilities: [
               AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY,
+              AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
               STRUCTURED_AGENT_SESSION_RUNTIME_CAPABILITY,
               CLAUDE_STRUCTURED_AGENT_SESSION_RUNTIME_CAPABILITY
             ]
diff --git a/src/main/runtime/rpc/methods/structured-agent-session-background-task-capability.test.ts b/src/main/runtime/rpc/methods/structured-agent-session-background-task-capability.test.ts
index cc5a7282080..4c2283efb52 100644
--- a/src/main/runtime/rpc/methods/structured-agent-session-background-task-capability.test.ts
+++ b/src/main/runtime/rpc/methods/structured-agent-session-background-task-capability.test.ts
@@ -1,6 +1,9 @@
 import { afterEach, beforeEach, describe, expect, it, vi } from 'vitest'
 import type { AgentSessionBackgroundTaskState } from '../../../../shared/agent-session-wire'
-import { AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY } from '../../../../shared/protocol-version'
+import {
+  AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
+  AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY
+} from '../../../../shared/protocol-version'
 import { remoteRuntimeClientCapabilities } from '../../../../shared/remote-runtime-client-capabilities'
 import type { AgentSessionSubscribeInput } from '../../../native-chat/agent-session-wire/structured-agent-session-subscribers'
 import {
@@ -24,6 +27,21 @@ const CURRENT_CLIENT = {
   ...STRUCTURED_CLIENT,
   clientCapabilities: remoteRuntimeClientCapabilities(STRUCTURED_CLIENT.clientCapabilities)
 }
+/** Understands a stopless roster, but predates per-row stoppability. */
+const STOP_ONLY_CLIENT = {
+  ...STRUCTURED_CLIENT,
+  clientCapabilities: remoteRuntimeClientCapabilities(STRUCTURED_CLIENT.clientCapabilities).filter(
+    (capability) => capability !== AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY
+  )
+}
+const MIXED_ROWS: AgentSessionBackgroundTaskState = {
+  state: 'monitoring',
+  supportsTaskStop: true,
+  tasks: [
+    { id: 'fore', kind: 'agent', stoppable: false },
+    { id: 'back', kind: 'agent' }
+  ]
+}
 
 describe('background-task stop capability at the RPC boundary', () => {
   it('advertises reader support on remote requests and subscriptions', () => {
@@ -93,6 +111,58 @@ describe('background-task stop capability at the RPC boundary', () => {
     }
   )
 
+  it('advertises row-stop support separately from stop support', () => {
+    // A client can advertise the stop capability and still predate `stoppable`,
+    // so the two must not be conflated.
+    expect(CURRENT_CLIENT.clientCapabilities).toContain(
+      AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY
+    )
+    expect(STOP_ONLY_CLIENT.clientCapabilities).not.toContain(
+      AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY
+    )
+  })
+
+  it.each([
+    ['row-stop reader', () => CURRENT_CLIENT, MIXED_ROWS],
+    [
+      'stop-only reader',
+      () => STOP_ONLY_CLIENT,
+      { state: 'monitoring', tasks: [MIXED_ROWS.tasks[1]] }
+    ],
+    ['in-process reader', () => undefined, MIXED_ROWS]
+  ] as const)('projects unstoppable rows for a %s', async (_label, client, expected) => {
+    hostCalls.history.mockReturnValue({
+      ok: true,
+      page: { items: [], backgroundTasks: MIXED_ROWS }
+    })
+    expect(
+      await call('agentSession.history', { sessionId: SESSION, direction: 'tail' }, client())
+    ).toMatchObject({ ok: true, result: { page: { backgroundTasks: expected } } })
+  })
+
+  it('hands a reader that predates the field no strip when every row is unstoppable', async () => {
+    // Its pre-feature view exactly: the host published no foreground rows at all.
+    const foregroundOnly = {
+      state: 'monitoring' as const,
+      supportsTaskStop: true,
+      tasks: [{ id: 'fore', kind: 'agent' as const, stoppable: false }]
+    }
+    hostCalls.history.mockReturnValue({
+      ok: true,
+      page: { items: [], backgroundTasks: foregroundOnly }
+    })
+    expect(
+      await call(
+        'agentSession.history',
+        { sessionId: SESSION, direction: 'tail' },
+        STOP_ONLY_CLIENT
+      )
+    ).toMatchObject({ ok: true, result: { page: { backgroundTasks: null } } })
+    expect(
+      await call('agentSession.history', { sessionId: SESSION, direction: 'tail' }, CURRENT_CLIENT)
+    ).toMatchObject({ ok: true, result: { page: { backgroundTasks: foregroundOnly } } })
+  })
+
   it('preserves legacy stoppable state for both readers', async () => {
     const stoppable = { state: 'monitoring', tasks: TASKS.tasks }
     hostCalls.history.mockReturnValue({ ok: true, page: { items: [], backgroundTasks: stoppable } })
diff --git a/src/main/runtime/rpc/methods/structured-agent-session-background-task-capability.ts b/src/main/runtime/rpc/methods/structured-agent-session-background-task-capability.ts
index 02afc5569bf..ae69de4a069 100644
--- a/src/main/runtime/rpc/methods/structured-agent-session-background-task-capability.ts
+++ b/src/main/runtime/rpc/methods/structured-agent-session-background-task-capability.ts
@@ -3,7 +3,10 @@ import type {
   AgentSessionHistoryResult,
   AgentSessionSubscribeEvent
 } from '../../../../shared/agent-session-wire'
-import { AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY } from '../../../../shared/protocol-version'
+import {
+  AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
+  AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY
+} from '../../../../shared/protocol-version'
 import type { RpcContext } from '../core'
 
 type BackgroundTaskReader = Pick<RpcContext, 'clientKind' | 'clientCapabilities'>
@@ -15,14 +18,37 @@ function supportsReadOnlyTasks(ctx: BackgroundTaskReader): boolean {
   )
 }
 
+function honoursRowStop(ctx: BackgroundTaskReader): boolean {
+  return (
+    ctx.clientKind === undefined ||
+    ctx.clientCapabilities?.includes(AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY) === true
+  )
+}
+
+/** A reader that predates `stoppable` draws a per-row stop on every row it is
+ *  handed, and the host cannot honour one on a row marked unstoppable — the
+ *  dead button the field exists to remove. The host publishing such rows at all
+ *  is new, so withholding them hands that reader exactly its pre-feature view;
+ *  a state whose every row is withheld becomes no strip, as it was. */
+function withoutUnstoppableRows(
+  state: AgentSessionBackgroundTaskState
+): AgentSessionBackgroundTaskState | null {
+  if (!state.tasks?.some((task) => task.stoppable === false)) {
+    return state
+  }
+  const tasks = state.tasks.filter((task) => task.stoppable !== false)
+  return tasks.length > 0 ? { ...state, tasks } : null
+}
+
 function projectState(
   state: AgentSessionBackgroundTaskState | null | undefined,
   ctx: BackgroundTaskReader
 ): AgentSessionBackgroundTaskState | null | undefined {
+  const rows = !state || honoursRowStop(ctx) ? state : withoutUnstoppableRows(state)
   // Legacy readers always offer a stop; retain their pre-producer empty strip.
-  return state?.supportsStopAll === false && !state.supportsTaskStop && !supportsReadOnlyTasks(ctx)
+  return rows?.supportsStopAll === false && !rows.supportsTaskStop && !supportsReadOnlyTasks(ctx)
     ? null
-    : state
+    : rows
 }
 
 export function projectBackgroundTaskHistory(
diff --git a/src/renderer/src/web/web-runtime-connection-frame-router.ts b/src/renderer/src/web/web-runtime-connection-frame-router.ts
index fe089be80ef..bb0d51b1380 100644
--- a/src/renderer/src/web/web-runtime-connection-frame-router.ts
+++ b/src/renderer/src/web/web-runtime-connection-frame-router.ts
@@ -1,6 +1,7 @@
 import type { RuntimeRpcResponse } from '../../../shared/runtime-rpc-envelope'
 import { isKeepaliveFrame } from '../../../shared/runtime-rpc-envelope'
 import {
+  AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
   AGENT_SESSION_BOUNDARY_RUNTIME_CAPABILITY,
   SESSION_TAB_CLOSE_INTENT_RUNTIME_CAPABILITY,
   SESSION_TABS_RETIREMENT_PROOF_DELTA_RUNTIME_CAPABILITY,
@@ -58,6 +59,7 @@ export async function routeWebRuntimeConnectionFrame(
           type: 'e2ee_auth',
           deviceToken: context.pairingToken,
           clientCapabilities: [
+            AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
             SESSION_TAB_CLOSE_INTENT_RUNTIME_CAPABILITY,
             SESSION_TABS_RETIREMENT_PROOF_DELTA_RUNTIME_CAPABILITY,
             AGENT_SESSION_BOUNDARY_RUNTIME_CAPABILITY,
diff --git a/src/shared/protocol-version.ts b/src/shared/protocol-version.ts
index 74317dd28c1..beeb37a5faa 100644
--- a/src/shared/protocol-version.ts
+++ b/src/shared/protocol-version.ts
@@ -166,6 +166,13 @@ export const AGENT_SESSION_REWIND_RUNTIME_CAPABILITY = 'agent-session.rewind.v1'
 // Readers must understand a monitoring roster with no available stop control.
 export const AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY =
   'agent-session.background-task-stop.v1' as const
+// Why: the host now publishes rows for work that is live inside a turn, and such
+// a row carries `stoppable: false` because no targeted stop can reach it. A
+// reader that predates the field draws a per-row Stop on every row it is given,
+// so it must be told apart from one that honours the field — and NOT by the
+// stop capability above, which a client can advertise while predating this.
+export const AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY =
+  'agent-session.background-task-row-stop.v1' as const
 // Why: adding kimi to RESUMABLE_TUI_AGENTS grows terminal.ensureAgentSession's enum, and an
 // older host answers the unknown member with invalid_argument — a code the launch fallback does
 // not retry on — so clients must probe before taking the host-authority path.
@@ -273,6 +280,7 @@ export const RUNTIME_CAPABILITIES = [
   AGENT_SESSION_STATUS_FEED_RUNTIME_CAPABILITY,
   AGENT_SESSION_REWIND_RUNTIME_CAPABILITY,
   AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY,
+  AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
   AGENT_SESSION_KIMI_RESUME_RUNTIME_CAPABILITY,
   FILE_MUTATION_OWNERSHIP_RUNTIME_CAPABILITY,
   GITHUB_MARK_PR_READY_RUNTIME_CAPABILITY,
diff --git a/src/shared/remote-runtime-client-capabilities.ts b/src/shared/remote-runtime-client-capabilities.ts
index 456f90e25ad..21e6bb145fa 100644
--- a/src/shared/remote-runtime-client-capabilities.ts
+++ b/src/shared/remote-runtime-client-capabilities.ts
@@ -1,4 +1,5 @@
 import {
+  AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
   AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY,
   AGENT_SESSION_BOUNDARY_RUNTIME_CAPABILITY,
   AUTOMATION_OWNER_FENCING_RUNTIME_CAPABILITY,
@@ -18,6 +19,7 @@ export function remoteRuntimeClientCapabilities(
   return Array.from(
     new Set([
       AGENT_SESSION_BACKGROUND_TASK_STOP_CAPABILITY,
+      AGENT_SESSION_BACKGROUND_TASK_ROW_STOP_CAPABILITY,
       SESSION_TAB_CLOSE_INTENT_RUNTIME_CAPABILITY,
       SESSION_TABS_AUTHORITATIVE_INVENTORY_RUNTIME_CAPABILITY,
       AGENT_SESSION_BOUNDARY_RUNTIME_CAPABILITY,