[Bug]: Cached thread rows keep claiming Working while their environment reconnects #25449

When an environment stops answering during a run, the client can show a reconnecting banner while its cached sidebar row still says Working with an increasing elapsed timer. This is observable even when Working shelf is enabled; a pinned thread remains in the pinned list by design.

Reproduction:

1. Start a remote thread and optionally pin it.
2. Interrupt the connection while its last received runtime is running.
3. Wait for the environment connection presentation to become reconnecting.
4. The sidebar continues to claim Working using cached runtime data.

Expected: distinguish unavailable connection from live work, suppress the live-work timer while disconnected, and restore the current run status after reconnect. Preserve run outcomes and pin/shelf placement; lost connectivity alone is not evidence of failure or completion. No elapsed-time cutoff is needed.

Scope: web/desktop sidebar and mobile thread-list status, using existing shared connection state. Investigating the server outage, recovery actions for orphaned threads, lineage redesign, and the separate mobile completion/shelf bug are outside this fix.

Confirmed on main 3ca62a5 with deterministic failing status-resolver tests for web and mobile. Prior related reports: #4852 (closed), #4589 (subscription loss while transport stays connected); closed prior attempts #8194 and #8935. #24901 concerns authoritative completion updates and is distinct.