It doesn't run https at all. This behaviour is also present on Maxthon though.
After analyzing the error myself further, I've followed a suggestion from https://stackoverflow.com/questions/23977690/setting-the-value-of-dataurl-exceeded-the-quota
I am also relaying it here for posterity in case anyone else stumbles over this; All I had to do was run
localStorage.clear();
inside the F12 Console window, and it seems to work fine now. Not sure what broke this in the first place, but at least it seems to work now, so this can be closed.