[Bug] Desktop app: Send button shows 'Executing' state when no task is running

Description

On the desktop app (latest version), the send button occasionally gets stuck in the "Executing" / in-progress state even when no task or background process is running. The input area appears blocked as if a task is still executing.

Refreshing the page resolves the issue immediately and the send button returns to normal.

Steps to Reproduce

1. Use the desktop app (latest version) on macOS
2. Have an active session with background wakeup/loop tasks
3. After a background task completes, the send button sometimes remains in "Executing" state
4. No task is actually running - the UI state is stale

Expected Behavior

Send button should return to normal idle state after all tasks complete.

Actual Behavior

Send button stays in "Executing" state, preventing message input until the page is refreshed.

Environment

• Platform: Desktop app (latest)
• OS: macOS

Related

This may be related to #97276 (duplicate response display) - both seem to be UI state synchronization issues after background task completion.