not really, a strong suspicion is enough to motivate assigning an extra paper and pen in person test to a student, and then you can fail them on that result.
That seems pretty unfair. Why not make the original test pen and paper then? (Or at least a typewriter, offline computer, etc - my handwriting is awful)
Originally the idea was that a take-home essay would allow students to work at their own pace, study their own way, and produce something interesting. But if more than half will just prompt an AI and learn nothing, then I agree you should proctor all assessments.