Reflexive differential nonvanishing conjecture

Let $X$ be a normal projective variety with klt singularities, and let $1\leq q\leq\dim X$. Write $\Omega^{[q]}_X$ for the sheaf of reflexive holomorphic differentials in degree $q$, and write $S^{[m]}\Omega^{[q]}_X$ for its reflexive $m$th symmetric power. A coherent sheaf is pseudoeffective in the sense of the paper's definition.

Reflexive differential nonvanishing conjecture. The sheaf $\Omega^{[q]}_X$ is pseudoeffective if and only if there exists a positive integer $m$ such that

$$H^0\left(X,S^{[m]}\Omega^{[q]}_X\right)\ne 0.$$

This extends the cotangent-bundle nonvanishing prediction from smooth projective manifolds to reflexive differentials on klt varieties. Its general status is not resolved in the supplied text.