The positive total scalar curvature conjecture

Let $X$ be a compact complex manifold in class $\mathcal C$, meaning that $X$ is bimeromorphic to a compact Kähler manifold. A Gauduchon metric is a Hermitian metric whose associated $(1,1)$-form satisfies $\partial\overline\partial\omega^{n-1}=0$; its total scalar curvature is the integral of its scalar curvature against the volume form. Positive total scalar curvature conjecture. $\kappa(X)=-\infty$ if and only if there exists a Gauduchon metric with positive total scalar curvature. The source says the necessary-condition direction is well known and proves that this conjecture is equivalent to the pseudo-effectivity/$\mathbb Q$-effectivity conjecture. Its general status remains open in the source.