I think if you look at the 2012 Harvard cheating scandal, it's clear that this isn't true. There, the professor presumed honest students, hundreds cheated, and no student reported.
One reason it did work is the students liked being trusted, and they did not like anyone that would threaten the system, and would turn them in.
BTW, that was 50 years ago. I have no information on how the honor system is fairing today.