New message badge on Mac will not clear (log out is the only thing that works)

I get messages, like all of us, and sometimes read and delete them on my iPhone, sometimes on one of my Macs. For a while, I had annoying problems with "New message badges" (the little red dot on the app icon on Mac) that would not go away, even if I had read and deleted the message on my phone. Then it got better, and stopped happening.

Uh oh, it's happening again. The only way I can find to get rid of that annoying badge is to log out of the User profile I'm in. But with two user profiles across three Macs, that six unneeded logout/login cycles each time this happens, which is not an acceptable solution.

I suppose it's tricky to track this stuff across the cloud and multiple devices using various system versions. But given the annoyance it causes -- I'm the kind of person who wants to clear those badges when possible -- is also high, so I hope Apple keeps working on this glitch.

Macs are all up-to-date in OS Sonoma, iPhone (where I originally deleted the message) is an iPhone 14 pro with up to date system (17.4.1).

A while ago, when this happened, someone here recommended having Siri read the "new" (but not really) message on my Mac to clear the badge. That does not work now (if it ever did): Siri just says I have no new messages, even though the little red badge with a number is right in front of me).