Schnell's conjecture for algebraic fiber spaces

Let $f\colon X\to Y$ be an algebraic fiber space between smooth projective varieties, and let $F$ be its general fiber with $\kappa(F)=0$. Suppose that there are an ample divisor $H$ on $Y$ and a positive integer $m_0$ such that

$$m_0K_X-f^*H$$

is pseudo-effective. Schnell's conjecture. Then

$$\kappa(X)=\dim Y.$$

This is the algebraic-fiber-space problem to which Schnell reduces the Campana–Peternell conjecture under non-vanishing. The paper proves it under additional hypotheses, including rigidity of the canonical class of the general fiber, but the conjecture itself remains open.