I have an idea of what might be the issue. I am using a normalized pressure distribution function. It scales with contact patch length. This simplifies the model, but makes the slope of the pressure distribution dependent on patch length, makes it softer.
Blue: short contact length
Red: longer contact length
Black: more realistic shape for longer contact length
I'll try implementing the piecewise (linear, quadratic) distribution with constant slope. It should result in increasing stiffness for longer contact lengths (I think...) and would be quite noticeable with the 4 tread model.