That seems like an easy solution, even if they just stored it for a few weeks.

Could be done for chatGPT but once an open source version is available this won't be possible.

If the whole motivation here is to detect the cheating student, most cheating students won’t simply copy and paste but will spend at least 5 - 15 min making modifications and writing some in their own language.

Policing cheating beyond punishing those who obviously are cheating in the worst ways is not as important as one might think. Cheating on highly competitive graduate school entrance exams is something to strictly police, but not an English writing assignment or a math word problem. It sounds corny, but the student in those cases really is just cheating themselves.

Any professor who talks to you in person or in class discussions or office hours, sees how you interact in group projects, and any employer who works with you on complex real-world problems chatGPT can’t solve will know very quickly that you (the cheater) don’t have a firm grasp of the material, prerequisites, or know-how to apply it, and the student or employee that does have that know-how and understanding gets the promotion, better Reference, has the better grades still (on average over all classes), and will interview much better for jobs and can speak intelligently about what they accomplished and solve problems on the spot on a white board, while the cheater fumbles and cannot pull out chatGPT during the interview, lol.