Clean up profile directory during New Identity

At CCCamp 2019 a cypherpunk approached me proposing that we try to clean up the profile directory as well as good as we can during New Identity.

I think I like the general idea of doing a more aggressive cleaning (from time to time) in the profile directory in particular if users are requesting a New Identity. I am not sure yet, though, whether we want to have this mechanism during New Identity as this one is usually more concerned with stuff that is risk exposing previous browsing sessions to web content. Which is clearly not the case for worries about the profile directory which are concerned with a local attacker.