feat(web): improve session sidebar with pinning, sorting, and unread indicators

Feature Request

The session list sidebar in the web UI currently has limited management capabilities. Several improvements would make it more practical for users managing multiple sessions.

Suggested Improvements

1. Pin / Favorite sessions

• Allow users to pin important sessions to the top of the list
• Pinned sessions should stay at the top regardless of activity

2. Customizable sort order

• Currently, sessions auto-sort by most recent activity (the session that receives a new message jumps to the top), which can feel disorienting
• Offer sorting options: by last activity, by creation date, alphabetical, or manual order
• Consider making the auto-reorder behavior optional, as it can be confusing when a session suddenly jumps to the top just because a message was sent

3. Unread message indicators

• Show a badge or dot on sessions that have new/unread messages
• Distinguish between "new assistant reply" and "no new activity"
• This is especially important when running multiple sessions in parallel — users need to know which sessions have new responses

4. Better activity sorting behavior

• The current "most recently active session goes to top" behavior can be disorienting
• Consider a less aggressive approach: highlight the session with new activity but keep the list order stable, or only reorder on user-initiated actions

Context

When working with HAPI across multiple sessions simultaneously, the sidebar is the primary navigation tool. These improvements would significantly help users manage and track their active sessions.

Start by locating the web UI session list/sidebar entry point and reading how session activity currently determines its order. Define the existing session state and ordering behavior before implementing pinning, sorting choices, stable ordering, and unread indicators; done means the sidebar supports the requested management options and clearly shows new activity.