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.polling.test.ts

@@ -97,7 +97,11 @@ function createManagerWithClient(clientOverrides: Record<string, unknown> = {}):
       ...clientOverrides,
     },
   }
-  return new BackgroundManager({ client, directory: tmpdir() } as unknown as PluginInput)
+  return new BackgroundManager(
+    { client, directory: tmpdir() } as unknown as PluginInput,
+    undefined,
+    { enableParentSessionNotifications: false },
+  )
 }
 
 describe("BackgroundManager verifySessionExists", () => {
@@ -201,6 +205,39 @@ describe("BackgroundManager pollRunningTasks", () => {
       //#then
       expect(task.status).toBe("completed")
     })
+
+    test("#when output was already observed from events #then it completes without fetching messages", async () => {
+      //#given
+      let messagesCallCount = 0
+      const manager = createManagerWithClient({
+        status: async () => ({ data: { "ses-idle-cached": { type: "idle" } } }),
+        messages: async () => {
+          messagesCallCount += 1
+          return {
+            data: [{
+              info: { role: "assistant", finish: "end_turn", id: "msg-2" },
+              parts: [{ type: "text", text: "done" }],
+            }],
+          }
+        },
+      })
+      const task = createRunningTask("ses-idle-cached")
+      injectTask(manager, task)
+
+      manager.handleEvent({
+        type: "message.part.updated",
+        properties: { sessionID: "ses-idle-cached", type: "text" },
+      })
+
+      //#when
+      const poll = (manager as unknown as { pollRunningTasks: () => Promise<void> }).pollRunningTasks
+      await poll.call(manager)
+      manager.shutdown()
+
+      //#then
+      expect(task.status).toBe("completed")
+      expect(messagesCallCount).toBe(0)
+    })
   })
 
   describe("#given a running task whose session status is busy", () => {