The semiampleness conjecture for K-trivial varieties

Let $X$ be a projective klt variety with $K_X \equiv 0$. Assume that $H^1(X, \mathcal{O}_X)=0$. Let $D$ be a nef divisor on $X$.

Semiampleness conjecture. Then $D$ is semiample.

This is a stronger expected form of abundance for K-trivial varieties. The source presents it as a conjecture and notes that only limited higher-dimensional cases are known.