Desktop app: embedded browser pane loses auth sessions (cookies) between app restarts

The desktop app's embedded browser pane doesn't persist cookie/session state across app restarts (and possibly across sessions). Practical impact: signing into GitHub in the pane to view private-repo links has to be redone essentially every sitting — for a workflow where the assistant links PRs and issues in a private org, that's several re-logins a day.

Repro:

1. In a session, open a link to a private GitHub repo in the embedded browser pane → GitHub serves 404 (unauthenticated, private repo).
2. Sign into GitHub in the pane → link renders.
3. Restart the app (or start a new session) and open another private-repo link → 404 again; the GitHub session is gone.

Expected: the pane's cookie jar persists like a normal browser profile (while staying isolated from the system browser — the isolation itself is good).

Environment: Claude Code 2.1.220, desktop app, macOS (Darwin 25.5.0).

Note: if non-persistence is deliberate (privacy/isolation policy), a small "this pane forgets sessions on restart" hint near the address bar would convert confusion into an informed choice — the current behavior reads as a bug even if it isn't one.