For an assignment completed at home, on a student's device using software of a student's choosing, there can essentially be no proof. If the situation you describe becomes common, it might make sense for a school to invest into a web-based text editor that capture keystrokes and user state and requiring students use that for at-home text-based assignments.
That or eliminating take-home writing assignments--we had plenty of in-class writing when I went to school.
According to an undergraduate student who babysits for our child, some students are literally screen recording the entire writing process, or even recording themselves writing at their computers as a defense against claims of using AI. I don't know how effective that defense is in practice.