Based on this info, I tried deleting <user-directory>/.gradle/caches/modules-2 and restarting Android Studio and it seemed to fix the PSI error, and it didn't seem to delete local history (though I don't know if anything else important was deleted).
Invalidating caches didn't work for me.
Only a complete re-install of AS and all relevant SDK, Gradle etc... did the trick.
I started by uninstalling AS.
Then I manually deleted the SDK folder.
I also deleted the .gradle and .android folders in my C:/Users/[my.user]/ folder.