A very expensive pressure transducer (and I don't mean AutoZone vs. Walmart tire gage) would show a difference between the loaded and unloaded tire. The unloaded tire has a bit higher volume -- it expands to the shape where pressure is mimimized. Any change in shape from this means a lower volume, thus slightly higher pressure. This is insignificant compared to the change in footprint that accompanies the deflection under load. Weight on tire [force] = Pressure [force/length^2]*Contact_Area [length^2] in any consistent units.

The US Olympic cycling team was sure they were going to win big because they had the technological edge, including the 'right' gasses in their tires. Their results said they should have spent the time training. The winners probably had ordinary air in their tires... but you can bet that the tires were pumped up that morning.