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 $\kappa (X,-K_X)\le \kappa (X_y,-K_{X_y})+\kappa (Y,-K_Y)$ whenever the $\mathbb{Q}$-linear series $\mathrm{\mid }-K_X{\mathrm{\mid }}_{\mathbb{Q}}$ has good singularities on $X_y$.