TUI shows stale background task count after tasks complete

Description

The TUI status bar continues to show background tasks after they have completed. The API reports all tasks as status: completed, but the UI still displays them in the task count.

Steps to Reproduce

1. Launch multiple background agents using the Task tool with run_in_background: true
2. Wait for all agents to complete
3. Verify completion via TaskOutput tool (all show status: completed)
4. Observe the TUI still shows "X background tasks" in the status bar

Expected Behavior

The background task counter should decrement to 0 (or the indicator should disappear) when all tasks complete.

Actual Behavior

The counter persists showing the original number of background tasks even after completion.

Screenshot

Status bar showing "6 background tasks" when all 10 checked tasks return status: completed:

▸▸ accept edits on · 6 background tasks

Environment

• Claude Code CLI
• Windows 11
• Multiple Task agents spawned for parallel library processing

Additional Context

Checked task IDs via TaskOutput tool:

• abddd68: completed
• a05b142: completed
• a945560: completed
• a9d8402: completed
• af2650e: completed
• aecbfe4: completed
• a47e04e: completed
• ae88896: completed
• acce070: completed
• ad16ecf: completed

All return status: completed but UI doesn't reflect this.