Generalized abundance for symplectic varieties

Let $X$ be a primitive symplectic variety, and let $L$ be a nef line bundle on $X$. A line bundle is semiample if some positive tensor power is generated by global sections.

Generalized abundance for symplectic varieties. Then $L$ is semiample.

This conjecture would produce a fibration from the linear system of a sufficiently divisible multiple of $L$. It is stated as a conjectural abundance result and remains open in the given generality.