Local First / Offline and Solid Login

I think that the problem that @jeffz say and that I’ve faced too is that the user is not really logout, and the prompt for choosing another webid should be accessible after logout.
I f I logout I should be able to choose another webid to login. The matter is that the old webid is still somewhere and we can’t switch to another. Should I clean something in localstorage to be able to use another webid ?