fix: prevent stale timeout from killing actively running background tasks

The stale detection was checking lastUpdate timestamps BEFORE
consulting session.status(), causing tasks to be unfairly killed
after 3 minutes even when the session was actively running
(e.g., during long tool executions or extended thinking).

Changes:
- Reorder pollRunningTasks to fetch session.status() before stale check
- Skip stale-kill entirely when session status is 'running'
- Port no-lastUpdate handling from task-poller.ts into manager.ts
  (previously manager silently skipped tasks without lastUpdate)
- Add sessionStatuses parameter to checkAndInterruptStaleTasks
- Add 7 new test cases covering session-status-aware stale detection

src/features/background-agent/task-poller.test.ts

@@ -136,6 +136,125 @@ describe("checkAndInterruptStaleTasks", () => {
     expect(task.error).toContain("no activity")
   })
 
+  it("should NOT interrupt task when session is running, even if lastUpdate exceeds stale timeout", async () => {
+    //#given — lastUpdate is 5min old but session is actively running
+    const task = createRunningTask({
+      startedAt: new Date(Date.now() - 300_000),
+      progress: {
+        toolCalls: 2,
+        lastUpdate: new Date(Date.now() - 300_000),
+      },
+    })
+
+    //#when — session status is "running"
+    await checkAndInterruptStaleTasks({
+      tasks: [task],
+      client: mockClient as never,
+      config: { staleTimeoutMs: 180_000 },
+      concurrencyManager: mockConcurrencyManager as never,
+      notifyParentSession: mockNotify,
+      sessionStatuses: { "ses-1": { type: "running" } },
+    })
+
+    //#then — task should survive because session is actively running
+    expect(task.status).toBe("running")
+  })
+
+  it("should interrupt task when session is idle and lastUpdate exceeds stale timeout", async () => {
+    //#given — lastUpdate is 5min old and session is idle
+    const task = createRunningTask({
+      startedAt: new Date(Date.now() - 300_000),
+      progress: {
+        toolCalls: 2,
+        lastUpdate: new Date(Date.now() - 300_000),
+      },
+    })
+
+    //#when — session status is "idle"
+    await checkAndInterruptStaleTasks({
+      tasks: [task],
+      client: mockClient as never,
+      config: { staleTimeoutMs: 180_000 },
+      concurrencyManager: mockConcurrencyManager as never,
+      notifyParentSession: mockNotify,
+      sessionStatuses: { "ses-1": { type: "idle" } },
+    })
+
+    //#then — task should be killed because session is idle with stale lastUpdate
+    expect(task.status).toBe("cancelled")
+    expect(task.error).toContain("Stale timeout")
+  })
+
+  it("should NOT interrupt running session task even with very old lastUpdate", async () => {
+    //#given — lastUpdate is 15min old, but session is still running
+    const task = createRunningTask({
+      startedAt: new Date(Date.now() - 900_000),
+      progress: {
+        toolCalls: 2,
+        lastUpdate: new Date(Date.now() - 900_000),
+      },
+    })
+
+    //#when — session running, lastUpdate far exceeds any timeout
+    await checkAndInterruptStaleTasks({
+      tasks: [task],
+      client: mockClient as never,
+      config: { staleTimeoutMs: 180_000, messageStalenessTimeoutMs: 600_000 },
+      concurrencyManager: mockConcurrencyManager as never,
+      notifyParentSession: mockNotify,
+      sessionStatuses: { "ses-1": { type: "running" } },
+    })
+
+    //#then — running sessions are NEVER stale-killed (babysitter + TTL prune handle these)
+    expect(task.status).toBe("running")
+  })
+
+  it("should NOT interrupt running session even with no progress (undefined lastUpdate)", async () => {
+    //#given — task has no progress at all, but session is running
+    const task = createRunningTask({
+      startedAt: new Date(Date.now() - 15 * 60 * 1000),
+      progress: undefined,
+    })
+
+    //#when — session is running
+    await checkAndInterruptStaleTasks({
+      tasks: [task],
+      client: mockClient as never,
+      config: { messageStalenessTimeoutMs: 600_000 },
+      concurrencyManager: mockConcurrencyManager as never,
+      notifyParentSession: mockNotify,
+      sessionStatuses: { "ses-1": { type: "running" } },
+    })
+
+    //#then — running sessions are NEVER killed, even without progress
+    expect(task.status).toBe("running")
+  })
+
+  it("should use default stale timeout when session status is unknown/missing", async () => {
+    //#given — lastUpdate exceeds stale timeout, session not in status map
+    const task = createRunningTask({
+      startedAt: new Date(Date.now() - 300_000),
+      progress: {
+        toolCalls: 1,
+        lastUpdate: new Date(Date.now() - 200_000),
+      },
+    })
+
+    //#when — empty sessionStatuses (session not found)
+    await checkAndInterruptStaleTasks({
+      tasks: [task],
+      client: mockClient as never,
+      config: { staleTimeoutMs: 180_000 },
+      concurrencyManager: mockConcurrencyManager as never,
+      notifyParentSession: mockNotify,
+      sessionStatuses: {},
+    })
+
+    //#then — unknown session treated as potentially stale, apply default timeout
+    expect(task.status).toBe("cancelled")
+    expect(task.error).toContain("Stale timeout")
+  })
+
   it("should release concurrency key when interrupting a never-updated task", async () => {
     //#given
     const releaseMock = mock(() => {})