I clear all notifications and it goes to zero, but after a reload it jumps back to 1 again. This keeps showing up on both desktop and phone, so I’m guessing something’s off with the sync.
I’ve had this when a group PM was unread — open Messages > All, mark everything read, then Shift+Refresh and the badge finally stayed gone. In my case it was a lingering system DM I’d never opened on mobile.