Non-vanishing conjecture for cotangent bundles

Let $X$ be a smooth projective variety, and let $1 \leq q \leq \dim X$. A vector bundle $\mathcal{E}$ on $X$ is pseudoeffective when the tautological line bundle $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ on the projectivization $\mathbb{P}(\mathcal{E})$ is pseudoeffective. If $\Omega_X^{q}$ is pseudoeffective, then the non-vanishing conjecture for cotangent bundles.

$$H^0(X, S^m\Omega_X^q) \neq 0$$

for some $m > 0$.

This conjecture generalizes the non-vanishing conjecture from canonical bundles to exterior powers of cotangent bundles. The paper proves it for isotrivial elliptic surfaces; together with the result of Hüring and Peternell, the question is completely solved for surfaces with Kodaira dimension at most $1$.