No, they cannot. If they could, it would say so on their "Extra Privileges" list, but I checked fizz's, and it says Extra privileges:
Voting on questions
Voting on answers
Voting posts down
Retagging any question
Editing any question
Editing any answer
Editing any comment
Editing posts silently
Selecting answer for any question
Viewing IPs of anonymous posts
Flagging posts
Approving or rejecting posts
Hiding or showing any post
Posting on user walls
Viewing the newest users page
Viewing the special users page
This means they can only delete any post on their own wall or any of their posts on any wall (like every user with over 30 points can do). So yes, for some godforsaken reason, Gʟɪɢᴜʀʀ IS active. Maybe he has e-mail notifications on, nobody knows for sure.
Nothing has changed since 2014 that PM didn't already address.