[FEATURE] Desktop sidebar (Group by: Project): keep running sessions visible in their project group, limit each group to ~7 rows, and count background-task completion as recent activity

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Thank you for the desktop app. I use it every day across several project folders, often with long-running background tasks in more than one project at the same time. With Group by: Project, it's easy to lose track of that work:

1. When the foreground turn ends but a background task keeps running, the session looks like any idle session in the sidebar. Only the conversation view shows that tasks are still running.
2. Recency is based on the last message. If a background task finishes two hours after the last message, the session stays buried and I don't notice the result.
3. Each project group lists every session, so the sidebar gets very long.
4. With work running in several projects, I have to remember which project I gave which task to.

Proposed Solution

1. Per-session status in the sidebar row, aggregated from all of the session's background tasks and using the same data the conversation view uses: needs attention, failed, running, queued, completed (not yet viewed), idle. If one of two tasks finishes, the row still shows running. A failure replaces the spinner. States are told apart by icon, shape and tooltip, not by color alone.
2. About 7 rows per project group by default, with a separate "Show N more" for each group. Running, queued, needs-attention, failed and completed-but-unviewed sessions are always shown and use up the 7 slots first. If more than 7 sessions need to be shown, all of them are shown. N is the exact number of hidden sessions.
3. Recency = latest real activity: the later of the last message and the background task's own timestamps (started, state changed, completed, failed). A session moves up when its task completes. Opening it clears only the "completed" indicator, not that timestamp, so it doesn't drop back down. Newer activity elsewhere pushes it down as usual.
4. A collapsed project group shows a small summary in its header (for example a spinner with a count, or "!"), without expanding the group automatically.
5. Pinned sessions show the same indicators. Pin behavior is unchanged.

Alternative Solutions

A separate "Running" section at the top (#98957) would also help. I'd still prefer sessions to stay in their project group, because where a session sits tells me which project the work belongs to. Group by: State groups running sessions together but loses the project view.

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

At 10:00 I start a long review in project A. The turn ends at 10:05, but a background task keeps running. I keep working in other sessions, and 20 new sessions are created in project A. At 12:00 the 10:00 session should still be visible near the top of project A with a running indicator. At 12:07 the task completes: the indicator changes to "completed" and the session moves to the top. At 12:10 I open it: the indicator goes away, but the session keeps its 12:07 position until newer activity happens elsewhere.

Additional Context

Windows 11, Claude desktop app 2.19675.0. Related: #98957, #98565, #83699.

Thank you for considering this!