Beltrametti–Sommese conjecture on effective strictly nef divisors

Let $X$ be a smooth projective variety and let $D$ be an effective strictly nef divisor on $X$. A divisor $D$ is strictly nef if $D\cdot C>0$ for every curve $C$ on $X$.

Beltrametti–Sommese conjecture. If $D-K_X$ is nef, then $D$ is ample.

This conjecture concerns when numerical positivity of an effective divisor, together with positivity relative to the canonical divisor, forces ampleness. The source presents it as a conjecture of Beltrametti and Sommese; no resolution is given in the supplied text.