[FEATURE] Desktop sidebar: surface running tasks and unread conversations instead of collapsing everything into project groups

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

After a recent Claude Desktop update, the left sidebar collapses all conversations under their project groups by default.

When I run multiple tasks in parallel across different conversations/windows (a core workflow with the Code tab), the sidebar gives no visible indication of:

• which conversations currently have a task running, and
• which conversations have unread/finished results since I last viewed them.

To find them I have to expand each project group one by one and scan manually. As a result it is very easy to completely forget about a running task — the running/unread state is effectively invisible in the sidebar.

Environment: Claude Desktop (macOS), Code tab, Max plan.

Proposed Solution

• Add a persistent "Active / Running" section pinned at the top of the sidebar that always lists conversations with in-progress tasks (with a spinner or status indicator), regardless of which project group they belong to.
• Show unread badges on conversations — and roll them up onto collapsed project groups — when there are new results or the task is waiting for user input since last viewed.
• Optionally: allow pinning conversations, and remember each project group's expanded/collapsed state.

The key point: running and unread conversations should be explicitly surfaced, instead of requiring the user to expand and scan each collapsed project group one by one.

Alternative Solutions

Currently I work around this by manually expanding each project group and clicking through conversations to check task status, or by keeping separate OS windows open for each running task. Both are error-prone — it is easy to lose track of a task that finished (or got stuck waiting for input) long ago. Most IDEs / chat apps solve this with a pinned "active" area and unread badges.

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

1. I start a long-running coding task in a conversation under project A.
2. While it runs, I switch to project B and start another task, then a third one under project C.
3. All project groups in the sidebar are collapsed; nothing indicates that A and B still have tasks running, or that C has finished and is waiting for my input.
4. Later I realize I completely forgot about the task in project A.
5. With a pinned "Running / Unread" section at the top of the sidebar, I would see all active and unread conversations at a glance and never lose track of parallel work.

Additional Context

This became noticeable after the recent Claude Desktop update where the sidebar started collapsing all conversations into project groups by default. Running multiple parallel tasks across projects is exactly the workflow the Code tab encourages, so the sidebar should make "running" and "unread" states first-class and always visible. Similar patterns: unread badges + pinned/active sections in Slack, and running-job indicators in CI dashboards.