If Gauthier is reelected without campaigning, I don't see why he should resign. After all he has done nothing wrong (unless he sometimes payed his young but legal partners. That would obviously change matters, a lot.)
Also, "Kerry Gauthier" sounds a lot like Mary Gauthier.
Whether, or not, Gauthiers has done something illegal is one question. Whether, or not, he has done something wrong is another question. The entire DFL machinery has pretty much signed onto the premise that a man in his fifties soliciting sex from a seventeen year-old is in fact "wrong."
I'm sure this an issue in which people can hold differing positions. If the DFL wants to endorse his reelection as the elected nominee whom has done nothing wrong, that would be their choice.