Mumford's nonvanishing conjecture for pseudoeffective canonical bundles

Let $X$ be a complex projective manifold, and let $K_X$ denote its canonical bundle. A line bundle is pseudoeffective when its numerical class lies in the closure of the cone of effective divisor classes. Write $\Omega^1_X$ for the cotangent bundle. Mumford's nonvanishing conjecture. If $K_X$ is pseudoeffective, then there exists a positive integer $m$ such that

$$H^0\big(X,(\Omega^1_X)^{\otimes m}\big)\neq 0.$$

This formulation is equivalent to Mumford's characterization because a projective manifold is uniruled if and only if its canonical bundle is not pseudoeffective. The conjecture is known in several cases, including when the numerical dimension of the canonical bundle is zero or maximal, but remains open in general.