I open a few pages from "hot network questions", then dismiss the inbox messages, then switch to some page.
Expected: the "1" counter disappears;
Actual: "1" counter stays until I click on it again or refresh the page.
Update: it seems to happen not all the time. I have just observed correctly disappearing red badge when I switch browser to a tab with the page.
Update 2: I have just switched to this very page and saw red "1" notification emerging. It was one I remember dismissing already...