The $(C_d)$ conjecture for almost strictly nef line bundles

Let $F$ be a projective manifold of dimension $d$ with Kodaira dimension $\kappa(F)=0$, and let $L$ be an almost strictly nef line bundle, meaning that $L$ is the pullback of a strictly nef line bundle under a surjective birational holomorphic map from $F$ to a normal projective variety. The $(C_d)$ conjecture. The divisor $K_F+tL$ is big for every real number $t>d+1$. This is introduced as an auxiliary problem for applying the Iitaka fibration; the supplied text does not state that it has been proved in general.