[Desktop] Sidebar session indicator keeps pulsing gray while session is idle (2.1.197)

Environment

• Claude Desktop app with Claude Code sessions, claude-code runtime 2.1.197
• macOS (Darwin 25.3.0), Apple Silicon

Bug

A session entry in the sidebar keeps pulsing/blinking gray as if the agent were actively working, even though the session is fully idle.

Verified on the affected machine while the indicator was pulsing:

• no background shell tasks, no monitors, no scheduled wakeups pending for that session
• no child processes (renders/servers) attached; the only processes referencing the session were the app's own runtime
• harness task list empty

A second session in the same project directory had a live process doing legitimate work at the time — possibly the activity indicator is attributed to the wrong sidebar row, or the pulse state fails to clear after background tasks complete.

Repro (as observed)

1. Run a long session that uses many run_in_background bash tasks + a ScheduledWakeup (all complete).
2. Have a second active session in the same project.
3. The first (idle) session's sidebar row keeps pulsing gray indefinitely; it survives idling for hours. User reports it persists.

Expected

Activity pulse only while that session is actually running work, clearing when idle.

Impact

Cosmetic but confusing/annoying — user can't tell which session is actually busy.