The abundance conjecture for compact complex manifolds

Let $X$ be a compact complex manifold in class $\mathcal C$, meaning that $X$ is bimeromorphic to a compact Kähler manifold. Let $K_X$ denote its canonical bundle. A line bundle is pseudo-effective if its class lies in the pseudo-effective cone, and $K_X$ is $\mathbb Q$-effective when some positive tensor power has a nonzero global section. Abundance conjecture.

$$K_X\text{ is pseudo-effective if and only if }K_X\text{ is }\mathbb Q\text{-effective}.$$

This is an abundance-type assertion relating positivity of the canonical bundle to the existence of pluricanonical sections. The source states that the necessary-condition direction is well known and proves that this conjecture is equivalent to the conjecture concerning positive total scalar curvature; its general status is not resolved there.