On the superadditivity of anticanonical Iitaka dimension

Given a fibration $f: X \to Y$ with normal general fibre $X_y$, over a field of any characteristic, we establish the Iitaka-type inequality $\kappa(X,-K_X) \leq \kappa(X_y,-K_{X_y})+\kappa(Y,-K_Y)$ whenever the $\mathbb{Q}$-linear series $|-K_X|_{\mathbb{Q}}$ has good singularities on $X_y$.