Obviously this question relies heavily on 'suspension of disbeief' because no orthodontist, now or ever, would put braces, nevermind wired on headgear, on anyone on the day of the prom. Unlessthe patient wanted that and timed it that well but that just did not happen.  In reality it is a purely theoretical question and all of you wishing one way or the other I can say with experience that the person in the headgear will not enjoy the party.  It was not the prom but the semi formal spring dance in 1971. I was 14, a freshman in high school with a four j-hook headgear and the j-hooks were affixed so I could not get them off.  This article was written by my orthodontis thttps://dereferer.me/?https://www.jco-online.com/archive/1974/12/679-directional-edgewise-orthodontic-approach-part-2/
This is the exact style of headgear I wore (and the same braces) https://dereferer.me/?https://www.jco-online.com/media/2894/679-jco-img-2.jpg

I had been wearing it full time for three weeks due to my record of lying and non-cooperation.  My orthodontist insisted on full time wear and my parents supported him.  It was humiliating enough wearing that contraption to school, I had no choice, but I was not going to a semi formal dance with it on.  My mother insisted I go and I refused.  I was still really pissed I was forced to wear it and in no mood to go social with it on, the braces were bad enough.  My mother got Karen's mother to force Karen to be my date and they dragged us to the dance, all dressed up, using threats and intimidation.  Karen was very nice, we knew and liked each other enough but was also there under duress for having snuck into an r-rated movie for which her mother promised she was going to hell and was permenantly grounded.  Hell wasnt open that day so she was forced to go to the dance with me instead.   Karen's mother was a chaperone so as much as we both wanted to leave, we were stuck there.  We both just sat in a dark corner and ignored the dance.  True story.

Given better circumstances Karen and I might have enjoyed each other and the dance.  We became good friends but never dated again, not that I could get close and kiss her anyway I had four wires sticking out of my mouth.  She still had her braces on and we got lots of jokes about locking braces and about being careful in thunderstorms.  Everyone else thought it was funny.  Not us.  Sorry to be such a buzz kill on the prom fantasy.