You mean the IDE in Chrome and the running app on Firefox?
I haven’t tried, but at this point I’m afraid it will do a new session only once, just like it does a new session only once with the incognito window (see Jay’s suggestion).
Waiting for a solution, I have added a Log table and I add to that table when I need to log something. Then I download the Log table to csv and browse it with Excel. It ain’t pretty (and I’m not marking this a solution, simply because it is not), but it allows me to work for now.