Session list: show status (running / waiting for input) and open PRs

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

### What I'd like

Two additions to the "Recents" session list in the Claude Code desktop app.

**1. A status dot per session**

Today the list is sorted by time only. There is no way to tell, at a glance,
whether a session is still running, has stopped, or is blocked on an answer
from me. I have to open each one.

Three states would already help a lot:
- running
- waiting for my input
- done / idle

The data is partly there already: `list_sessions` exposes `isRunning`. What's
missing is "the last turn was a question to the user". I realise that one is
the hard case — it will be wrong sometimes. Even shipping just
running/idle would be an improvement.

**2. A filter or badge for sessions with an open PR**

This one is a hard fact rather than a guess, so it should be reliable.
`list_sessions` already returns `prNumber` and `prState`.

My concrete case: a PR touching order execution sat open for 16 days across
sessions. Nothing in the UI surfaced it. A badge on the session row, or a
"has open PR" filter, would have caught it on day one.

### Why it matters

I run several sessions in parallel across two projects. The cost of losing
track isn't cosmetic — in my case it's unmerged changes to money-handling
code drifting further from main every day.

### Environment

Claude Code desktop app, Windows 11.

### Proposed Solution

### Alternative Solutions

### What I'd like

Two additions to the "Recents" session list in the Claude Code desktop app.

**1. A status dot per session**

Today the list is sorted by time only. There is no way to tell, at a glance,
whether a session is still running, has stopped, or is blocked on an answer
from me. I have to open each one.

Three states would already help a lot:
- running
- waiting for my input
- done / idle

The data is partly there already: `list_sessions` exposes `isRunning`. What's
missing is "the last turn was a question to the user". I realise that one is
the hard case — it will be wrong sometimes. Even shipping just
running/idle would be an improvement.

**2. A filter or badge for sessions with an open PR**

This one is a hard fact rather than a guess, so it should be reliable.
`list_sessions` already returns `prNumber` and `prState`.

My concrete case: a PR touching order execution sat open for 16 days across
sessions. Nothing in the UI surfaced it. A badge on the session row, or a
"has open PR" filter, would have caught it on day one.

### Why it matters

I run several sessions in parallel across two projects. The cost of losing
track isn't cosmetic — in my case it's unmerged changes to money-handling
code drifting further from main every day.

### Environment

Claude Code desktop app, Windows 11.

### Priority

Critical - Blocking my work

### Feature Category

CLI commands and flags

### Use Case Example

_No response_

### Additional Context

_No response_

Start by tracing the `list_sessions` entry point and the desktop app's "Recents" session-list UI. Check how `isRunning`, `prNumber`, and `prState` reach each session row, then define the status and open-PR presentation. Done means the list visibly distinguishes running or idle sessions and reliably surfaces sessions with open PRs.

Rédigé par le modèle d'indexation à partir du texte de l'issue.