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.
That, or the uni shall give me a separate machine to write on, only for that purpose.
Well yes, in-person proctored is the gold standard. For those who can’t or won’t go in person, something invasive is really the only alternative to entirely exam-based scoring.