GPT-5.6 Sol: background subagent result is lost after parent ends waiting turn

**Issue type: Bug / possible regression**

- Copilot Chat Extension Version: 0.57.0
- VS Code Version: 1.129.1 (commit 8a7abeba6e03ea3af87bfbce9a1b7e48fed567b8, x64)
- OS Version: Windows 10 Pro, build 26200
- Feature: Agent mode / Autopilot Preview / asynchronous background subagents
- Selected model: GPT-5.6 Sol
- Logs: Persistent agent debug logging is enabled. The session records repeated reinitialization entries; no subagent completion or subagent error is recorded. On the next turn, querying the task returns `Agent not found`. I can provide sanitized logs privately if needed.

This appears to be a regression of #322230. I have repeatedly observed it while using GPT-5.6 Sol. I have not established whether the model itself is causal, so the model association should be treated as a correlation.

## Steps to Reproduce

1. Start an Agent/Autopilot session using GPT-5.6 Sol.
2. Launch a custom subagent as an asynchronous background task. In my case it is a Playwright video-capture agent expected to run for several minutes.
3. The parent responds that it is waiting for the subagent and the UI marks the parent task as `Finished`.
4. Wait for the background task or send the next message to the same conversation.
5. Try to inspect the task/subagent result.

## Actual result

- The background task is no longer present and task lookup returns `Agent not found`.
- No completion or failure result is surfaced to the parent conversation.
- The work is effectively orphaned, and the subagent must be launched again.
- This has happened repeatedly with different long-running subagent attempts.

## Expected result

- The parent session should remain subscribed to the asynchronous subagent until it completes, or the completed result should be persisted and delivered on the next turn.
- A conversation/session reinitialization should not discard the background task handle or its result.
- If the task is cancelled, the UI should surface an explicit cancellation/error rather than silently losing it.

## Additional information

- The behavior closely matches #322230, which was closed after async-subagent fixes.
- These settings are enabled, but do not prevent the loss:
- `chat.requestQueuing.defaultAction: queue`
- `chat.notifyWindowOnResponseReceived: always`
- `github.copilot.chat.agentDebugLog.enabled: true`
- `github.copilot.chat.agentDebugLog.fileLogging.enabled: true`
- `chat.agent.maxRequests` is set to 50000, so this does not appear to be the request limit.
- I could not find a supported setting for background-subagent persistence or timeout.