diff --git a/lib/__tests__/sw-push-delivery.test.ts b/lib/__tests__/sw-push-delivery.test.ts
index fe4b57ee4..005258e7d 100644
--- a/lib/__tests__/sw-push-delivery.test.ts
+++ b/lib/__tests__/sw-push-delivery.test.ts
@@ -295,4 +295,26 @@ describe('nothing new to say', () => {
   });
 });
 
+describe('a push naming several accounts, one server slow', () => {
+  it('does not make the others wait out its deadline', async () => {
+    const started: string[] = [];
+    let release = () => {};
+    const slow = new Promise<void>((resolve) => { release = resolve; });
+    const fetchSlow = vi.fn(async (input: string) => {
+      const url = new URL(input, 'https://mail.example.com');
+      const account = url.searchParams.get('accountId') ?? '';
+      started.push(account);
+      if (account === 'slow') await slow;
+      return { ok: true, json: async () => ({ unreadTotal: 1, slot: 0, email: mail(`m-${account}`) }) };
+    });
+    const w = loadWorker(() => null, { globals: { fetch: fetchSlow } });
+    const done = w.push({ kind: 'jmap-state-change', accountId: 'slow', changed: { slow: { EmailDelivery: 's' }, fast: { EmailDelivery: 's' } } });
+    await vi.waitFor(() => expect(started).toEqual(['slow', 'fast']));
+    await vi.waitFor(() => expect(w.shade.has('bulwark-mail:fast')).toBe(true));
+    release();
+    await done;
+    expect(w.shade.size).toBe(2);
+  });
+});
+
 // -- end of cases --
diff --git a/lib/__tests__/sw-push-subscription-change.test.ts b/lib/__tests__/sw-push-subscription-change.test.ts
index 813a8fa69..50eb98290 100644
--- a/lib/__tests__/sw-push-subscription-change.test.ts
+++ b/lib/__tests__/sw-push-subscription-change.test.ts
@@ -7,7 +7,7 @@ import { describe, expect, it, vi } from 'vitest';
 // in a VM with a minimal ServiceWorkerGlobalScope and fire the event.
 const SW_SOURCE = readFileSync(join(process.cwd(), 'public', 'sw.js'), 'utf8');
 
-function loadWorker(tabs: { postMessage: (msg: unknown) => void }[], registrations?: unknown) {
+function loadWorker(tabs: { postMessage: (msg: unknown) => void }[], registrations?: unknown, config: { noThirdPartyTraffic?: boolean } | null = {}) {
   const listeners = new Map<string, (event: unknown) => void>();
   const newSubscription = { toJSON: () => ({ endpoint: 'https://push.example/new', keys: { p256dh: 'p', auth: 'a' } }) };
   const subscribe = vi.fn(async (_options: { userVisibleOnly?: boolean; applicationServerKey?: Uint8Array }) => newSubscription);
@@ -23,8 +23,12 @@ function loadWorker(tabs: { postMessage: (msg: unknown) => void }[], registratio
     self,
     atob,
     caches: { open: async () => ({ match: async () => (registrations ? { json: async () => registrations } : undefined) }) },
-    fetch: vi.fn(async (url: string, init: { body: string }) => {
-      relayCalls.push({ url, body: JSON.parse(init.body) });
+    fetch: vi.fn(async (url: string, init?: { body: string }) => {
+      if (url.endsWith('/api/config')) {
+        if (!config) throw new TypeError('offline');
+        return { ok: true, json: async () => config };
+      }
+      relayCalls.push({ url, body: JSON.parse(init!.body) });
       return { ok: true };
     }),
     URL, URLSearchParams, Response, Date, console, Uint8Array,
@@ -69,6 +73,18 @@ describe('service worker pushsubscriptionchange', () => {
     expect(relayCalls).toHaveLength(2);
   });
 
+  it('calls no relay while third-party traffic is off: the page moves it', async () => {
+    const { fire, relayCalls, newSubscription } = loadWorker([], REGISTRATIONS, { noThirdPartyTraffic: true });
+    await fire('pushsubscriptionchange', { oldSubscription: null, newSubscription });
+    expect(relayCalls).toEqual([]);
+  });
+
+  it('calls no relay when it cannot find out', async () => {
+    const { fire, relayCalls, newSubscription } = loadWorker([], REGISTRATIONS, null);
+    await fire('pushsubscriptionchange', { oldSubscription: null, newSubscription });
+    expect(relayCalls).toEqual([]);
+  });
+
   it('calls no relay for a browser where push was never on', async () => {
     const { fire, relayCalls, newSubscription } = loadWorker([]);
     await fire('pushsubscriptionchange', { oldSubscription: null, newSubscription });
diff --git a/public/sw.js b/public/sw.js
index 76e1a767b..47cde4b71 100644
--- a/public/sw.js
+++ b/public/sw.js
@@ -603,7 +603,9 @@ async function handlePush(event) {
   const withNewMail = emailIds.length > 0 || !changed
     ? []
     : Object.keys(changed).filter((id) => changed[id] && typeof changed[id] === "object" && "EmailDelivery" in changed[id]);
-  for (const accountId of withNewMail.length > 0 ? withNewMail : [named]) {
+  // Side by side: each has its own preview deadline, and Safari wants the
+  // notifications within 30 s of the push.
+  await Promise.all((withNewMail.length > 0 ? withNewMail : [named]).map(async (accountId) => {
     try {
       await notifyAccount(accountId, emailIds, accountLabel);
     } catch (_) {
@@ -617,7 +619,7 @@ async function handlePush(event) {
         data: { kind: "mail-list", accountId },
       });
     }
-  }
+  }));
 }
 
 async function notifyAccount(accountId, emailIds, accountLabel) {
@@ -1000,6 +1002,18 @@ async function readRelayRegistrations() {
   }
 }
 
+/** No-third-party mode may have come on since the page left a relay here. Unknown counts as no. */
+async function mayCallRelays() {
+  try {
+    const res = await fetch(`${BASE_PATH}/api/config`, { cache: "no-store" });
+    if (!res.ok) return false;
+    const config = await res.json();
+    return config.noThirdPartyTraffic !== true;
+  } catch (_) {
+    return false;
+  }
+}
+
 function applicationServerKeyOf(base64Url) {
   const padded = base64Url + "=".repeat((4 - (base64Url.length % 4)) % 4);
   const raw = atob(padded.replace(/-/g, "+").replace(/_/g, "/"));
@@ -1031,7 +1045,7 @@ async function handlePushSubscriptionChange(event) {
       }
     }
   }
-  if (subscription && registrations && typeof subscription.toJSON === "function") {
+  if (subscription && registrations && typeof subscription.toJSON === "function" && (await mayCallRelays())) {
     const json = subscription.toJSON();
     for (const entry of Object.values(registrations.entries)) {
       try {