In old UI, Jira comments related notification are displayed in header section of the page. Those notification will show the count of new comments and that count will be reset only on next day.
Need to implement this functionality in latest UI with better enhancements and in an alternate way to reset the count. Also planning to show the browser notification (based on configuration) even if Jira Assist is not open.
If anyone is interested in this feature then +1 this ticket with suggestion if any.