tlwebadm doesn't get any tab icon in either Safari or Epiphany. In Safari you get some autogenerated icon (the first letter of the domain), but in Epiphany you get nothing at all.
It it unknown why things fail at this point. One theory is that it is the authentication that messes things up for some reason.
Tested with Safari 16 on macOS 13, and Epiphany 41.4 on Fedora 35.
It does indeed seem to have something to do with auth. I can see that it gets 401 responses:
> 2022-11-10 15:36:10 INFO tlwebadm: [::ffff:10.48.2.194] 'GET /images/icons/icon.ico HTTP/1.1' 401 -
This is however after I've authenticated, as otherwise it wouldn't be able to load the page that gives it the URL for the icon. So it seems it doesn't pass along the authentication information to whatever loads the icon.
It generally seems like Webkit (and historically Chrome, judging from Google hits) handles the icon completely separately from loading the rest of the page. It isn't shown in the network inspector, and it isn't affected by the "Disable Caches" setting.
Filed a bug for this with WebKit:
This bug also applies to every browser on iOS.
Also seems to be an issue with Firefox on Android.