The uniruledness conjecture for compact Kähler manifolds

Let $X$ be a connected compact Kähler manifold.

Uniruledness conjecture. $X$ is uniruled, meaning covered by rational curves, if and only if

$$H^0(X,K_X^{\otimes k})=\{0\}$$

for all $k>0$.

This is presented as a consequence of the Abundance Conjecture and gives a pluricanonical criterion for uniruledness in the compact Kähler setting.