That makes more sense
There's at least three of these add-ons that work very similarly. Here's one I wrote, but have a search around for some others that may be suitable:
https://xenforo.com/community/resources/live-update.274/
That will periodically check for new messages, and alerts, and if it finds any, it will update the alert/conversation
bulls (bubbles
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) and also display the count in the title of the browser tab or optionally as a red icon appearing over your favicon.
It works on any page, not just the index page.