His schedule should be something like this at school, and they ARE REQUIRED by most state laws to have any children under kindergarten age to take around a 1.5 hour nap, can be a little more and can be a little less. But kids under kindergarten age fall into child care age laws and they're different than elementary school age kids laws.
Up around 7 and go to child care/pre-school/etc...
Around 11 eat lunch
11:30noon lay down for nap
2pm up from nap, snack, ready for home.
around 6 dinner.
play time, family time, what life is about for the family.
Bedtime 8-9
In my opinion too many parents think "have dinner, kids off to bed, my time". I'm not saying that's what your thoughts are but is there any real reason kids can't be up later and spend time with mom and dad?
In all my years of child care and being a parent and having 7 grandchildren and friends with 4-8 kids I find that little kids don't need 10-12 hours of sleep at night. No one is wired that way.
The TOTAL numbers of RECOMMENDED sleep for kids includes their naps and rest times. If he's sleeping at all during the day that counts against his night time hours. He can't go to bed at 7pm and be expected to sleep past 5am. That's 10 hours. Kids will sleep maybe 9 at his age.