The vanishing criterion for non-pseudo-effectivity of the canonical divisor

Let $X$ be a klt projective variety over $\C$. Suppose that

$$H^i(X,L)=0$$

for $i>0$ and every semi-ample line bundle $L$ on $X$. The vanishing criterion conjecture asserts that $K_X$ is not pseudo-effective.

If true, this would provide a cohomological criterion forcing a klt projective variety outside the pseudo-effective range for the canonical divisor, with consequences for the birational geometry used in the paper's rational-chain-connectedness results. The source says that the conjecture would allow the non-vanishing assumption in one of its main results to be dropped; no resolution is given.