Active-chats section at top of sidebar, like pinned chats

[davidtaylor6130]

Summary

Similar to the pinned-chats section at the top of the sidebar, add an active chats section that surfaces chats that need attention.

Requested behavior

A section at the top of the sidebar (near/above the pinned chats section) listing chats that are:

1. Actively processing — a turn is currently running in that chat
2. Finished but unseen — processing completed but the user has not yet viewed the output

• Viewing/selecting a finished chat clears its "unseen" state and removes it from the section
• A processing chat that completes moves from state 1 to state 2 (stays visible until looked at)
• Section hides entirely when empty (no dead space in the sidebar — see Remove empty pin space and tighten sidebar hierarchy #150)

Motivation

Today the only way to know which background chats are still running or finished and waiting is to click into each one. An active-chats section gives an at-a-glance view of where attention is needed.

Notes

• Related but distinct from Visible background panes remain stale after completion #271 (stale background panes after completion) — that one is a rendering bug, this is a new sidebar section.
• Scope is the top sidebar section only. No separate bottom "all chats" grouping.