[FEATURE] Desktop Code tab: a Settle button, so the sidebar works as an inbox #109182

T3 Code has a Settle button on every thread. One click marks the thread done and tucks it out of the list, and sending it a message brings it back. That makes the sidebar an inbox: what's in view is only what I'm still working on.

In the Claude desktop app I can't tell which sessions are taken care of and which I'm still in the middle of. Archive is the closest thing, but there's no place in the sidebar to see archived sessions (#102116), and it isn't one click from the session.

What I'd like:

• A Settle button in every session, next to the model picker or in the header, and in the row's right-click menu
• Settled sessions go to one collapsed group at the bottom of the sidebar, below the date list
• Sending a message to a settled session unsettles it
• The same for sessions started from the phone (Remote Control), and the same button in the mobile app

I built this as a mod for now: a footer button that files the session into a "Settled ✓" section. Three things the mod can't do are why it should be built in: a custom section can't sit below the date list, a mod can't draw anything in a Remote Control session, and move_sessions refuses those sessions (#104002).

Related: #102116, #107422, #106459.