Shh! Don't give the answer away before the Wednesday 3pm ET deadline.
Edit: The room is round, so the spot where the letters wrap must be important otherwise it could have been a single wall with several doors. Also, the letters are *between* the doors. So this points to the exit door being being between M and H, but why? Reread the question and focus on opening and out; they seem to be key.
A: When marked with "OUT" the door between M and H forms a word for an opening — "MOUTH".This is my theory as to the intended answer, but I'll have to wait to Sunday to have it officially confirmed.