fix(background-agent): skip terminal tasks during stale pruning

Prevent TTL pruning from deleting terminal tasks before delayed notification cleanup runs.

Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)

Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>

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

@@ -1,4 +1,5 @@
-import { describe, it, expect, mock } from "bun:test"
+declare const require: (name: string) => any
+const { describe, it, expect, mock } = require("bun:test")
 
 import { checkAndInterruptStaleTasks, pruneStaleTasksAndNotifications } from "./task-poller"
 import type { BackgroundTask } from "./types"
@@ -447,4 +448,38 @@ describe("pruneStaleTasksAndNotifications", () => {
     //#then
     expect(pruned).toContain("old-task")
   })
+
+  it("should skip terminal tasks even when they exceeded TTL", () => {
+    //#given
+    const tasks = new Map<string, BackgroundTask>()
+    const oldStartedAt = new Date(Date.now() - 31 * 60 * 1000)
+    const terminalStatuses: BackgroundTask["status"][] = ["completed", "error", "cancelled", "interrupt"]
+
+    for (const status of terminalStatuses) {
+      tasks.set(status, {
+        id: status,
+        parentSessionID: "parent",
+        parentMessageID: "msg",
+        description: status,
+        prompt: status,
+        agent: "explore",
+        status,
+        startedAt: oldStartedAt,
+        completedAt: new Date(),
+      })
+    }
+
+    const pruned: string[] = []
+
+    //#when
+    pruneStaleTasksAndNotifications({
+      tasks,
+      notifications: new Map<string, BackgroundTask[]>(),
+      onTaskPruned: (taskId) => pruned.push(taskId),
+    })
+
+    //#then
+    expect(pruned).toEqual([])
+    expect(Array.from(tasks.keys())).toEqual(terminalStatuses)
+  })
 })

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

@@ -13,6 +13,13 @@ import {
 } from "./constants"
 import { removeTaskToastTracking } from "./remove-task-toast-tracking"
 
+const TERMINAL_TASK_STATUSES = new Set<BackgroundTask["status"]>([
+  "completed",
+  "error",
+  "cancelled",
+  "interrupt",
+])
+
 export function pruneStaleTasksAndNotifications(args: {
   tasks: Map<string, BackgroundTask>
   notifications: Map<string, BackgroundTask[]>
@@ -22,6 +29,8 @@ export function pruneStaleTasksAndNotifications(args: {
   const now = Date.now()
 
   for (const [taskId, task] of tasks.entries()) {
+    if (TERMINAL_TASK_STATUSES.has(task.status)) continue
+
     const timestamp = task.status === "pending"
       ? task.queuedAt?.getTime()
       : task.startedAt?.getTime()