I discovered today that if you skip the tutorial (which levels you to level 11) neither buddy gets the level 10 buddy-up chest.
It's not that big of a deal for the mentor buddy - they only lose, I think, 20 battle solutions - but the recruit buddy loses out on a permanent loot pet.
I've contacted support, but considering the weekend, I probably won't hear back until Monday. But as a PSA, make sure recruits don't skip the tutorial.