The best information I have been able to find is that sidewall stiffness - that is the stiffness of the sidewall by itself - plays little role in rolling resistance.
However, inflation pressure does. Inflation pressure will stiffen a sidewall and in fact is the primary component when you want to compare sidewall stiffness inflated. For that reason, ride quality is very much tied to inflation pressure and only slightly so to the actual stiffness of the sidewall by itself. That is not to say you can not feel the difference in stiffness between tires. It is to say that inflation pressure has more of a difference.
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