fix(background-agent): cache observed output for completion checks

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

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

src/features/background-agent/manager.test.ts

@@ -5020,6 +5020,66 @@ describe("BackgroundManager.handleEvent - non-tool event lastUpdate", () => {
     //#then - task should still be running (delta event refreshed lastUpdate)
     expect(task.status).toBe("running")
   })
+
+  test("should complete idle task without fetching messages after output event was observed", async () => {
+    //#given - a running task with observed output from message part events
+    let messagesCallCount = 0
+    let todoCallCount = 0
+    const sessionID = "session-output-cached-idle"
+    const client = {
+      session: {
+        prompt: async () => ({}),
+        promptAsync: async () => ({}),
+        abort: async () => ({}),
+        messages: async () => {
+          messagesCallCount += 1
+          return {
+            data: [
+              {
+                info: { role: "assistant" },
+                parts: [{ type: "text", text: "ok" }],
+              },
+            ],
+          }
+        },
+        todo: async () => {
+          todoCallCount += 1
+          return { data: [] }
+        },
+      },
+    }
+    const manager = new BackgroundManager({ client, directory: tmpdir() } as unknown as PluginInput)
+    stubNotifyParentSession(manager)
+
+    const task: BackgroundTask = {
+      id: "task-output-cached-idle",
+      sessionID,
+      parentSessionID: "parent-session",
+      parentMessageID: "msg-1",
+      description: "idle cached output task",
+      prompt: "test",
+      agent: "explore",
+      status: "running",
+      startedAt: new Date(Date.now() - (MIN_IDLE_TIME_MS + 10)),
+    }
+    getTaskMap(manager).set(task.id, task)
+
+    manager.handleEvent({
+      type: "message.part.updated",
+      properties: { sessionID, type: "text" },
+    })
+
+    //#when - session.idle fires after output event was already observed
+    manager.handleEvent({ type: "session.idle", properties: { sessionID } })
+
+    //#then - task completes without refetching session.messages
+    await new Promise((resolve) => setTimeout(resolve, 10))
+    expect(task.status).toBe("completed")
+    expect(messagesCallCount).toBe(0)
+    expect(todoCallCount).toBe(1)
+
+    manager.shutdown()
+  })
 })
 
 describe("BackgroundManager regression fixes - resume and aborted notification", () => {