After a fair amount of rapid “W” converting via the Proselytize button, it bugged out and broke. No cool-down. Just stays grayed out (as seen in screenshot), such that it can never work for that monk.
There were many monks, and this was just one as far as I could tell. But you can see he has it disabled. I waited the cool-down time just to make sure, but it made no difference. Thanks for reading.