There's more to potty training readiness than just staying dry through a nap, each child is ready when THEY are ready, and readiness can be taught.
I would keep him in the pull-ups, or better yet cloth trainers with plastic pants over them for his nap. Pull-ups ARE diapers, kids know this and use them as such. They're made to absorb and wick the wetness away from their bodies, so a little one feels comfortable peeing in them.
Each and every child is different, even when dealing with multiples. My guy trained by 27 months, even nap and night dryness. But that was because HE chose to get up and use the potty at naps and night, I was content to put him in a diaper still. I truly believe it was because I never used pull-ups, when we were done with diapers I put him in cloth trainers, then "cool" underwear within the week, a diaper only the first 2 nights. He felt the wetness and didn't like it and chose to stay dry on his own. It's a process, some are just ready before others.