[BUG] Desktop session list: open-PR marker leaks onto every session sharing the repo directory

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

In the desktop app's session list, opening a PR from one session paints the PR indicator on
**every other session in the same repository**, including sessions that never touched git and
sessions that were last active days before the PR existed.

I checked the underlying session metadata (the same data the sidebar renders) and the marked
sessions have no PR linkage at all:

- Exactly **one** session in that project carries PR state: `prNumber: `, `prState: "OPEN"`,
on the feature branch that PR was opened from.
- The other marked sessions have **no `prNumber` field whatsoever**. Several were last active
3–5 days before that PR was opened.
- None of those sessions carry a `branch` field either. Only worktree-backed sessions do
(`cwd: …/.claude/worktrees/`, `branch: claude/`).
- The repository's currently checked-out branch is that same feature branch — the PR's head ref.

So the indicator is not derived from each session's own metadata. Per the diagnostic work in
#59647, it is a **project-scoped cache that is never invalidated**: after checking the repo
back out to `main` (clean tree, PR branch checked out nowhere in that directory), the badge
still renders, and clears only after a full restart of the desktop app.

### What Should Happen?

The indicator should be scoped to the session it belongs to, and rendered from that session's
own recorded `prNumber` / `prState` / `branch`. A session with no PR linkage should show
nothing. When the branch state changes, the indicator should be re-derived without needing an
app restart.

### Error Messages/Logs

```shell
(no error output — this is a rendering/state bug)
```

### Steps to Reproduce

1. Run several sessions in one repository over a few days, all from the repo root (no
worktrees), so they share a single working directory.
2. From one of those sessions, create a branch and open a PR (`gh pr create`). Leave that
branch checked out.
3. Open the session sidebar filtered to that project.
→ **Every** session in the project now shows the PR indicator, not just the one that
opened the PR. Clicking the indicator on an older session opens the newer session's PR
(also reported in #66505).
4. Now `git checkout main` in that directory. Confirm `git branch --show-current` → `main`
and a clean tree; the PR branch is checked out nowhere in that directory.
→ The indicator is **still shown**.
5. Quit and relaunch the desktop app.
→ Only now does it clear.

### Claude Model

Opus

### Is this a regression?

I don't know

### Last Working Version

/

### Claude Code Version

2.1.234 (Claude Code)

### Platform

Anthropic API

### Operating System

macOS

### Terminal/Shell

Other

### Additional Information

**This is the desktop app, not the CLI** — the Terminal/Shell field above is not applicable.

- Claude desktop app: **1.34493.1**
- Claude Code: **2.1.234**
- macOS **26.5.1**, Apple silicon

**Prior reports, both closed without a fix:**

| Issue | Title | How it closed |
|---|---|---|
| #66505 | PR indicators incorrectly appearing on all sessions in same repository | Auto-closed as a duplicate of #59647, then locked |
| #59647 | PR status badge is shown in every session's composer, not only the session that created the PR | Closed `not planned` by the stale bot after inactivity |

#59647 also carries the most useful diagnosis in its comments: per-session worktrees stop
sessions from sharing a `HEAD`, but they do **not** fix this — the badge persists after the
branch is gone from the directory and clears only on app restart, so the defect is cache
invalidation and scoping, not `cwd` resolution. It also notes the leak spans two surfaces: the
composer PR badge and the session-list indicator.

**Why file again rather than comment:** both issues are closed, and #66505 is locked with a
notice asking that a new issue be filed referencing it.

**Still present on current versions.** The earlier reports were at 2.1.133 and 2.1.168; this
reproduces on desktop 1.34493.1 / Claude Code 2.1.234.

**Possible interaction with #86838.** That issue (PR badges resetting to "open" after an app
update) has a comment saying it was fixed in 1.34493.1 — the exact version I'm on. Worth
checking whether the badge-recovery fix there re-broadened the badge's scope. I have not
verified any link; flagging it only so triage can rule it in or out.

**Impact:** with many sessions per repository, the indicator marks a dozen sessions as having
open PRs when one does, and its links point at the wrong session's PR — which removes the
signal it exists to provide.

Start with the desktop session sidebar and composer PR-badge paths, then inspect how session metadata and the project-scoped PR cache are read and invalidated; #59647 describes the related diagnostic. Reproduce with several sessions sharing one repository, create and then check out of a PR branch, and verify that only the session with its own prNumber/prState/branch shows the indicator without requiring an app restart.

Written by the indexing model from the issue text.