Quiz setting to close quiz when computer is closed?

This is our 1st year on Brightspace and I'm an instructional designer asking this for faculty in a department where class sizes are 200 or so students. Their concerns are that a few students start taking a Brightspace quiz (exam) in the classroom, and then at some point appear to be finished and leave the room. However, these students haven't yet submitted it, and some of them reopen their laptops elsewhere and complete the quiz by cheating (accessing other materials or perhaps collaborating). I suggested setting the quiz to start/stop at specific times, and requiring everyone to stay put for the duration. But, if someone HAS to leave, they can come forward and prove they've submitted the quiz. Or just give their names, and the time they leave can be recorded and checked against their quiz submission time. Faculty in the department weren't excited about these suggestions. (Too much work or that the lecture halls have too many doors to monitor.) Does anyone have other suggestions?