At 4K the frequency of any excited wave mode is pretty low, so I would think that the linear approximation is pretty good in this range.
However, you might note that the dispersion need not be isotropic. (Need to know the dispersion omega(vec k), it isnt' just omega(|k|). this might need to be accounted for.
More likely the problem has to do with how the system has been measured --- it is notoriously difficult to measure heat capacity of one system sitting on top of another (how do you tell if heat is leaking from one to the other?). So they may very well have measured it wrong.