Stale background task indicators persist in UI after completion

Description

Background task/agent indicators continue showing "running" status in the UI after the tasks have completed. The user sees multiple tool runs that "aren't doing anything" - ghost entries that never clear.

Steps to Reproduce

1. Launch multiple background agents using Task tool with run_in_background: true
2. Wait for agents to complete (confirmed via TaskOutput showing status: completed)
3. Observe that the UI still shows tasks as running

Expected Behavior

Task indicators should clear or show "completed" status when background agents finish.

Actual Behavior

Task indicators persist showing running/active status even after:

• TaskOutput confirms status: completed
• ps aux shows no related processes running
• Work output has been successfully retrieved

Environment

• Claude Code version: 2.1.12
• Platform: WSL2 (Linux 6.6.87.2-microsoft-standard-WSL2)
• Context: Using parallel background agents for Neo4j migration work

Additional Context

User observed 4 stale task indicators that persisted across conversation continuation (claude --continue).