  We show that for a surjective, separable morphism f of smooth projective
varieties over an algebraically closed field of positive characteristic such
that $f_* \mathcal{O}_X = \mathcal{O}_Y$ subadditivity of Kodaira dimension
holds, provided the base is of general type and the Hasse-Witt matrix of the
geometric general fiber is not nilpotent.
