[FEATURE] Desktop: see the status of many sessions at once, without filtering

Related to #88020, but a different problem. That one is about arranging panes. This one is about seeing status without opening anything.

What I do

I run about 30 Claude Code sessions in parallel, every day, for work.

The problem

Today the desktop app gives me two ways to see session state:

• Split view, which shows two sessions: "To view two sessions at once, hold Cmd on macOS or Ctrl on Windows and click a session in the sidebar."
• Sidebar filters: "filter sessions by status, project, or environment"

Both are pull. I have to decide to look, then filter, then read the list. With 30 sessions I do that many times an hour, and most of the time the answer is "nothing needs me yet".

What I want is push: the state of all sessions visible at the same time, without any action from me. Then I only move when something changes.

What I would like

A compact status view of all sessions in one screen. It does not need to show the transcripts. It needs one row or one tile per session with:

• which state it is in (working / waiting on you / done)
• the project or worktree
• how long it has been in that state

Colour is enough for the state. Reading text is what I am trying to avoid.

Why the count matters

At 2 to 5 sessions, cycling with Ctrl+Tab is fine. At 30, the cost is not switching, it is not knowing which one to switch to. The sidebar can list them, but a list of 30 rows still has to be read.

On the time part

"How long it has been in that state" is the one I miss most. A session that has been working for 6 hours and a session that started 10 minutes ago look the same today. Once I had an agent rewriting and rerunning the same tests for most of a week and I did not notice, because it kept reporting progress. Elapsed time would have shown it in one glance.