Thanks Patrick, in this case it was another extension not resolving the promise expected from a listener it added for `webRequest.onAuthRequired`. So, our listener for the same auth challenge couldn't complete the authentication. The purpose of our extension is for customers to connect to our proxy network, so the bug stops our customers being able to do use our service.
Luckily the other extension devs are being cooperative and agree they'll work on a fix, but if that had not been the case then it seems we'd have been pretty powerless to do much about it or even do a release that recognises the issue and warns the user somehow. It seems like a bit of a loose end that extensions can break each other so easily and there aren't many tools to detect or deal with it, or automatic warnings based on the permissions each is using or something...