Strictly nef anticanonical bundles under smooth morphisms

Let $f:X\rightarrow Y$ be a smooth morphism between projective manifolds. The anticanonical bundles are denoted by $K_X^{-1}$ and $K_Y^{-1}$.

Direct-image conjecture for strict nefness. If $K_X^{-1}$ is strictly nef, then $K_Y^{-1}$ is strictly nef.

The conjecture extends the known preservation of ampleness of the anticanonical bundle under smooth morphisms. The paper notes that it would follow from the Campana–Peternell conjecture and gives a partial result: under the hypothesis, $Y$ is rationally connected.