Background task counter not decremented after TaskOutput acknowledgment or session compaction

Bug Description

The status bar shows "3 background tasks" even after:

1. All background tasks have completed (confirmed via task-notifications)
2. TaskOutput was called on completed tasks (returned status: completed)
3. Task output files were cleaned up

The counter persists and cannot be cleared.

Steps to Reproduce

1. Start background tasks using run_in_background: true on Bash tool
2. Wait for tasks to complete (receive task-notification with status: completed)
3. Call TaskOutput on the completed task IDs - returns success with status: completed
4. Observe status bar still shows the same number of background tasks

Additionally:

1. If session runs out of context and gets compacted/summarized
2. Background task counter persists but task IDs are lost
3. Orphaned tasks can never be acknowledged

Expected Behavior

• Calling TaskOutput on a completed task should decrement the background task counter
• Session compaction should either preserve task IDs or reset the counter

Actual Behavior

• TaskOutput returns success but counter stays the same
• After session compaction, tasks become orphaned (counter persists, task IDs lost)

Environment

• Claude Code CLI
• Model: Opus 4.5
• Custom statusline enabled (but issue is with Claude Code's internal task tracking, not custom script)

Screenshot

Status bar showing "3 background tasks" after all tasks completed and acknowledged:

• Tasks bb5c918 and bf41fab both returned status: completed from TaskOutput
• Third task likely orphaned from pre-compaction session

Workaround

Currently none known except restarting the Claude Code session.