Updating Posit Assistant can fail if multiple RStudio windows have it active

### System details

RStudio Edition : Desktop or Server
RStudio Version : 2026.04.0
OS Version : any (but particularly bad on Windows)
R Version : any

If there are multiple RStudio sessions running with the Posit Assistant pane open in each, and one of those panes gives the option to update Posit Assistant, the update may fail because the files are in use (especially on Windows which locks executable files while they are running).

The instance where the update is happening knows to shut down Posit Assistant before doing the update but it doesn't check if there are other RStudio instances and notify them to shut down as well.

Even if the update doesn't fail (e.g. on Linux/Mac) we end up in a weird state where running code doesn't match the code on disk in those secondary instances, so having the chat shutdown and restart in all windows would be ideal.

Reproduce the Posit Assistant update with the pane open in multiple RStudio Desktop or Server sessions, especially on Windows, and trace the update flow from the instance offering the update. Check how other sessions are notified and how Assistant is shut down and restarted. Done means every active pane reaches a consistent state after the update without file-in-use failures.

Written by the indexing model from the issue text.