You can easily try jourself. 1) Have Firefox open 2) Start a root terminal (or a normal one and then "sudo su" 3) sudo -u normaluser firefox --new-window example.com
It will refuse to start.
It will refuse to start.
Do the following: sudo -EHs su, then as root sudo -EH -u {username} firefox --new-window
I'll leave to you the exercise of learning what the flags do, and implicitly why it fails in your case.