He does. He recommends getting as many as you can with good form on the last set of the day as long as it doesn't inhibit progress on the following weeks (and it's not the deload week).
He talks about it somewhere in the book but without scanning more I can't find the full thing but this is from the FAQ portion (pg 100):

So week 1: 5x65%, 5x75%, Max Repsx85%
Week 2: 3x70%, 3x80%, Max Repsx90%
Week 3: 5x75%, 3x85%, Max Repsx95%
Week 4 (Deload): 5x40%, 5x50% and 5x60% - No max reps this week at all
And in interviews since he's even stated when it might be acceptable to double the cycle progression. If you can get 12 reps week 1, 8 reps week 2 and 5 reps or more on week 3 then he says it's okay to double the advancement. So 10 lbs for upper body (instead of the usual 5) and 20 lbs for lower body (instead of the usual 10).
And yeah, I'm friends with a personal trainer at my gym and he's always telling me the same thing about 5/3/1. That it's not enough volume or weight to be effective. I can't do anything but shrug and say "it works for me". I hate those moments and I wish there was research (maybe there is somewhere? but I don't know about it) that I could use to back it up. But I stopped caring about convincing people about what works. Most don't care about trying anything outside their comfort zone anyways. I understand it's different for you since it's for your class though. Sucks I can't be of more help
Edit: Shit. I found it. Page 23. The section called "The Last Set"