So, the good news is that since last Friday, all of my 4 stations seem to appear steadily online on their respective stats pages. No more spurious offline alerts. Nice!
The bad news is that there is still a new bug in the UI, both on the website and from the iOS mobile app, where only the last registered station (i.e. only 1 out of 4) is listed while I'm logged in. When I manually access one of the unlisted stations, all of them sometimes reappear in the menu, but not always. I wasn't able to figure out the bug pattern quite yet.
Nevertheless, I'm glad that the false alerts are gone, the remaining bug is rather minor.
Cheers and good week to all aviation fans,
Matt