test(agent-hooks): pin that leadStopWithLiveSubagents is never persisted

The strip in serializeStatusFile was the one part of the gate with no test.
Rehydrating the flag would make the notification gate suppress the first real
completion after every restart, which is the spam #4375 reports.

src/main/agent-hooks/server.test.ts

@@ -6597,6 +6597,32 @@ describe('Last-status persistence', () => {
     }
   })
 
+  it('never persists leadStopWithLiveSubagents, so a restart cannot re-suppress a completion', async () => {
+    const server = new AgentHookServer()
+    await server.start({ env: 'production', userDataPath })
+    try {
+      await postHookEvent(server, buildBody({ hook_event_name: 'SessionStart' }), '/hook/codex')
+      await postHookEvent(
+        server,
+        buildBody({
+          hook_event_name: 'SubagentStart',
+          agent_id: '11111111-2222-4333-8444-555555555555',
+          agent_type: 'explore'
+        }),
+        '/hook/codex'
+      )
+      await postHookEvent(server, buildBody({ hook_event_name: 'Stop' }), '/hook/codex')
+      server.flushStatusPersistSync()
+      const persisted = JSON.parse(readFileSync(lastStatusPath(), 'utf8'))
+      // Why: the flag describes one live hook delivery. Rehydrating it would make the
+      // notification gate suppress the first real completion after every restart (#4375).
+      expect(persisted.entries[PANE].payload.state).toBe('done')
+      expect(persisted.entries[PANE].payload).not.toHaveProperty('leadStopWithLiveSubagents')
+    } finally {
+      server.stop()
+    }
+  })
+
   it('treats a corrupt file as empty hydration without throwing', async () => {
     mkdirSync(join(userDataPath, 'agent-hooks'), { recursive: true })
     writeFileSync(lastStatusPath(), 'not-json{{', 'utf8')