Background tasks panel: completed tasks stay listed as Running and dismissals do not persist

Environment

• Claude Code 2.1.174 (desktop app)
• macOS 26.5.1 (Apple Silicon)

Summary

Two related problems with the Background tasks panel in long sessions:

1. Background tasks that finished hours ago are still listed under "Running" with a live elapsed timer. I had entries showing 6h49m that I could confirm were dead: their processes were gone from ps, their output files had stopped growing, and the agent had already consumed their results and moved on. Only a handful of genuinely live watchers existed at the OS level, but the panel showed a dozen "Running" rows.

1. Dismissing entries does not stick. I closed the stale rows with the control on each card, switched away from the window, and when I came back every one of them had reappeared. It looks like the close action only hides the row in the current view, and any re-render reads the full session task list again.

Steps to reproduce

1. Run a long session that spawns plenty of background work (Bash with run_in_background, background agents, monitors).
2. Let a number of them complete naturally over a few hours.
3. Open the Background tasks panel: completed entries still show as Running with growing timers.
4. Dismiss some entries, switch to another app or window, come back: the dismissed entries are back.

Expected

• Tasks whose process has exited move out of Running (or get a Completed state) without needing manual cleanup.
• Dismissing an entry persists for at least the lifetime of the session.

Notes

The stale entries seem to be cosmetic only. I checked ps and there were no leaked processes behind them, so this is a display-state issue rather than a resource leak. Both problems together make the panel hard to trust in long sessions: you cannot tell what is actually running, and you cannot clean it up.