Campana–Peternell reformulation for algebraic fiber spaces with Kodaira dimension zero fibers

Let $f:X\to Y$ be an algebraic fiber space between smooth projective varieties, and suppose that the general fiber $F$ satisfies $\kappa(F)=0$. Let $H$ be an ample divisor on $Y$. Campana–Peternell reformulation. If $m_0K_X-f^*H$ is pseudo-effective for some $m_0\geq 1$, then $\kappa(X)=\dim Y$. The paper presents this as an equivalent formulation under the non-vanishing conjecture, but the supplied text does not establish its resolution status.