Desktop: PR tracker bars are stale, cannot be dismissed, and sometimes show `main → main`

Three related issues with the PR/branch tracker panel that sits above the message composer in the Claude Code desktop app.

1. No way to dismiss stale PR tracker bars

The bars accumulate over time and there's no affordance to remove individual entries:

• Right-click does nothing.
• App restart does not clear them (verified after a fresh restart for an app update).
• Starting a new session via /clear has no effect.
• The three PR-related settings toggles (auto-create, auto-fix, auto-archive sessions) are unrelated — none of them control panel visibility or entry dismissal.

Expected: a way to dismiss an individual bar (e.g. right-click → dismiss, or a hover-×), and/or a setting to auto-hide entries whose PRs are merged or closed.

2. Stale state — merged PRs displayed as open

A PR that has already been merged on GitHub continues to render in the panel with live "CI" status and a "View PR" button, as if still open.

Concrete case: a repo where gh pr list --state all confirmed all 7 PRs were merged still showed all 2 branches in the panel as open, with CI dropdowns and View PR buttons. Multiple app restarts did not refresh the state.

<img width="2412" height="780" alt="Image" src="https://github.com/user-attachments/assets/71738224-fffa-4ce3-97ae-cce28c55128d" />

Expected: PR state should refresh on app start (and ideally on window focus), so merged PRs either disappear or visibly transition to a "merged" badge.

3. Pulling latest into main is sometimes visualized as a PR main → main

After a routine git pull on the local main branch, an entry sometimes appears in the panel showing main ← main, as if a PR were being opened from main to itself.

Expected: updates to main (e.g. fast-forward pulls) should not surface as PR tracker rows. Only feature branches with an associated open PR (or a credible "needs PR" state) belong here.