I'm going to say no. Sidewall stiffness shouldn't matter. Here's my reasoning. Say you have a 10.5 inch tread width contacting the ground. This contact - for sake of argument is 5" long. When you air down, you aren't increasing that width measurement (because you're not driving with the sidewalls on the ground) you're increasing the length measurement to say 12" or longer. This gives the more surface area which is beneficial off-road. So even if you have stiff sidewalls, you're still going to increase the length of the contact patch, and the width will be the same as when aired up.
I'd also like to clear up a mis-conception about airing down too. I've heard many times where people comment about how the gearing affects change and such when you air down. They say...well, I'm decreasing my tire radius, so it's like running a smaller tire. NO, it's not. Your vehicle doesn't care how tall your tire is...what matters is the circumference (length around tire tread). This measurement doesn't change.
Hope all this makes sense.
