I am having what I think is a similar issue. I am set up on two Teams sites, and many times a day I flip back and forth between the two. Lately every time I switch over I stop getting updates (any chat messages, reactions, posts) on the desktop. Teams still works on my mobile phone, so I see the notifications there and that's how I know I am not getting updates on my computer. Every time I have to quit and restart teams, but then as soon as I flip to another client site the same issue starts again. I tried uninstalling and reinstalling Teams desktop and it still did not help.