Kollár's non-uniruledness conjecture for general very ample divisors

Let $X$ be a smooth projective variety and $|H|$ a very ample linear system on $X$ such that $K_X+H$ is effective.

Kollár's conjecture. A general member $H\in |H|$ is not uniruled.

The preceding proposition establishes a sharp degree bound for hypersurface sections in the case of linear subspaces, while this conjecture proposes non-uniruledness under the broader adjoint-effectivity condition. The source presents it as a strongest variant that could be true and does not state a resolution.