Thank you very much for joining the discussion and sharing your experience 
I’ve had exactly the same issue, except that I was not logged out when it happened. Still, the new message appeared without any badge on the chat icon—I only noticed it after manually opening the chat section.
I’ve installed the DiscourseHub app on my phone just for testing purposes, but I don’t actively use it. The issue I described occurred on Windows, using the Edge browser.
Happy to provide more details if needed.