Desktop app: sidebar "Running" indicator stays pulsing on idle sessions (esp. after interrupting turns)

Environment: Claude Code desktop app 2.1.3, macOS 26.5.1 (Apple Silicon)

Symptom: Sessions in the sidebar show a pulsing dot with a "Running" tooltip even when nothing is running. Right now ~10 pinned/recent sessions are all pulsing simultaneously. In an affected session, TaskList returns "No tasks found," there is no in-flight turn, and all background tasks/monitors have long since completed — yet the indicator keeps pulsing indefinitely (hours).

Suspected trigger: frequently interrupting turns (Esc). Interrupted turns appear to leave the session's running-state flag set. Sessions where background tasks (bash watchers / monitors) completed normally may also fail to clear.

Expected: the running indicator clears when a turn completes or is interrupted and no tracked background work remains. Ideally "Running" should be derived from actual state (in-flight request or live task) rather than a sticky flag.

Impact: with many sessions the sidebar becomes a wall of pulsing dots, making it impossible to see which session is genuinely working — the indicator loses all signal.

Workaround: restarting the app clears all stale indicators, confirming the underlying sessions were idle.