First, NASCAR (and Goodyear) don't even allow that low a pressure, as it is unsafe. They probably start at 15-25 lb. depending on the track and the tire location on the car.
Second, the inner liners are only used on Superspeedway tires, not short track tires. They are inflated to about 60 lbs using a separate fill valve.
Lastly, there are actually very few plies in race tires, as they tend to be rather light weight. The NASCAR wheel assemblies are heavy because of the mandatory steel wheels, which have to be very strong to support the high side loads of 3600 lb. cars.
We tried nitrogen in our road race tires, and found the it was a royal pain to get enough of the air out of the tire so that the pressure rise of the nitrogen was predictable; it was easier to just stick with air, which was predictable.
Marty