On subadditivity of Kodaira dimension in positive characteristic

We show that for a surjective, separable morphism f of smooth projective varieties over an algebraically closed field of positive characteristic such that $f_{\ast }{\mathcal{O}}_X={\mathcal{O}}_Y$ 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.