How much does tire sidewall stiffness contribute to the force holding the car up? I know that can vary a lot. Some cars have run-flats that won't sag when they lose pressure (or won't sag as much, anyway). Low profile tires probably transfer more of the load through the sidewall than doughnut tires. And of course the contribution from the sidewalls (as a percent) will change as tire pressure increases.
I have some data on this for a few different tyres but I don't think I can legally quote it here :(