For sites using the favicon to indicate a new message / event, Falkon often doesn't update the icon in the tabs. I usually see the "new message" icon stuck forever. Maybe it's only updating the icon once and it breaks afterwards?
This was already reported and I didn't yet receive a good testing site for this (what I tested worked fine).
jfi: I seem to be able to reliably reproduce it with thelounge: https://github.com/thelounge/lounge
I can reproduce it with https://demo.thelounge.chat/
https://bugreports.qt.io/browse/QTBUG-64967