How (super) rough is the glassy phase of a crystalline surface with a disordered substrate?

We discuss the behaviour of a crystalline surface with a disordered substrate. We focus on the possible existence of a super-rough glassy phase, with height-height correlation functions which vary as the square logarithm of the distance. With numerical simulations we establish the presence of such a behaviour, that does not seem to be connected to finite-size effects. We comment on the variational approach, and suggest that a more general extension of the method could be needed to fully explain the behaviour of the model.