[A3] Add status reconciliation fallback for remote prompts #1567

Parent: #1564

Context

The app sets session_status = 'busy' optimistically on prompt submit with no timeout. If SSE events never arrive (fan-in gap, network issue), "Thinking" persists forever.

Change

After setting optimistic 'busy,' start a fallback timer. If no SSE event resolves the status within a timeout (30s), poll the session detail endpoint to check actual execution status and reconcile. Cancel the timer when the SSE event arrives normally.

Acceptance criteria

• A reconciliation timer starts on prompt submit
• If the SSE resolution event arrives, the timer is cancelled
• If the timer fires (30s default), a poll of the session endpoint reconciles the status
• The timer is cleaned up on session navigation / unmount
• New test: simulated SSE gap triggers fallback reconciliation

Files

• packages/app-bundle/overlay/packages/app/src/components/prompt-input/submit.ts
• packages/app-bundle/overlay/packages/app/src/context/server-session.ts