[FEATURE]: Surface sessions needing input on the existing Desktop home screen #54922

Related proposals

This overlaps with #53851 (a broader sessions view grouped by state) and #53757 (attention-first ordering in Web). This request proposes a smaller change to the existing Desktop home screen, without a new sidebar or sessions view.

Problem

The home screen lists sessions by date, but does not show which are working or waiting for me. With several sessions open, I have to enter them individually to find a pending permission or question.

Proposal

Keep the existing projects, search, and session navigation. Add:

• A Needs input section above the date-grouped list for sessions with an outstanding permission or question. Show the type on each row.
• A compact Working indicator for sessions actively running.
• An understated Idle indicator for sessions with neither active work nor outstanding input.
• A count beside the Needs input heading.

Within each section, retain the existing recency order. Avoid moving a row while the user is interacting with it. Clicking a Needs input row should open that session so the request can be answered.

For the initial version, this could be limited to the selected project and these three states. Errors, pins, filters, cross-project aggregation, and “ready for review” can be considered separately.

Correctness

Outstanding requests should appear after reopening the app, not only when they arrive while the home screen is open. A resolved permission or question should promptly leave Needs input. If a session is both running and awaiting input, Needs input takes precedence.

Mockup

Would maintainers find this narrow home-screen treatment useful, or prefer that it be considered entirely within #53851?