[BUG] Switching between accounts in Claude Desktop appears to pause running tasks on the other account

### Description
I have two separate paid Claude accounts and use both from the same Claude Desktop app. When I switch from account A to account B, tasks that were actively running on account A (Cowork background sessions) appear to pause — progress stops until I switch back to account A.

### Expected behavior
Per the Cowork documentation, sessions in the cloud should keep running in the background regardless of what's open in the app ("open the session from any surface to check its progress"). Since I'm paying for two separate subscriptions, I'd expect to be able to run active work on both accounts at the same time without one blocking the other.

### Actual behavior
Switching the active account in the desktop app seems to drop the live connection to the previous account's session, which either pauses the task or at least stops it from being visibly updated/driven until I switch back.

### Context
This is likely related to the lack of native multi-account support tracked in #18435 — the app doesn't maintain two independent authenticated sessions at once, so switching acts more like a sign-out/sign-in than a true account switch.

### Ask
- Confirm whether cloud-run Cowork tasks actually continue executing server-side during an account switch, or whether they are genuinely paused client-side.
- If they are paused: prioritize true concurrent multi-account support (or at minimum, don't interrupt in-flight tasks when switching the active account in the UI).

### Environment
- Claude Desktop app
- Two active paid accounts, switching within the same app instance

Start by reproducing the account switch in Claude Desktop with two paid accounts and an active Cowork session, then compare observed behavior with the Cowork documentation and related issue #18435. Done means establishing whether the session continues server-side and, if it does not, defining a way to switch accounts without interrupting in-flight work.

索引モデルが issue の本文から書いたものです。