They wear their straps under their lip because if they get into a fight if the strap is under their chin then the suspect could take advantage of it and grab onto their hat possibly choking them. So if they wear it under their lip then they wont be able to do that, instead the hat will just be pulled off.
“State police regulations indicate that the strap is to be worn (quote) extending across the point of the chin. The public information officer tells me sometimes, during the course of duty, the strap may end up creeping under the trooper’s lip.” How long does it take to adjust a chin strap to the regulation position; maybe two seconds (if you’re not eating a donut)? I suspect the wear position is more about comfort than about creeping up in the course of duty.