This is a good idea.

Ideally, the notification system would notify you without page reload in a useful way. Like "[New PM] <Username> responded to <Convo title>". And maybe they'd accumulate in a tray/sidebar that you can check while reading other things. But I wasn't sure what that would actually look like and that feels too far away to think about.

A persistent sidebar is a good middle ground since the system is so non-realtime right now. Maybe it could show your latest 5 convos and 5 subscriptions in a small stack, and regular forum usage would do the work of "refreshing" the list as you go do other things.