Mumford's conjecture that varieties of Kodaira dimension minus infinity are uniruled

Let $X$ be a smooth projective variety, and let $\kappa^h(X)$ denote its holomorphic Kodaira dimension. Mumford's conjecture. If

$$\kappa^h(X)=-\infty,$$

then $X$ is uniruled. This is the converse to the fact that uniruled manifolds have holomorphic Kodaira dimension $-\infty$. It is known for projective $3$-folds; the source states that in general it follows from the Abundance conjecture, so the general case remains open.