Schnell's algebraic fiber-space non-vanishing conjecture

Let $f:X\to Y$ be an algebraic fiber space, meaning that $X$ and $Y$ are smooth projective varieties and $f$ is surjective with connected fibers. Let $F$ be a generic fiber satisfying $\kappa(F)\geq 0$, and let $H$ be an ample divisor on $Y$. Schnell's algebraic fiber-space conjecture. If $mK_X-f^*H$ is pseudo-effective for some $m\geq 1$, then $mK_X-f^*H$ becomes effective for $m$ sufficiently large and divisible. This is the conjecture proposed by Schnell whose equivalence with the Campana–Peternell conjecture and the non-vanishing conjecture is discussed in the paper; the supplied text does not independently provide a resolution status.