The optimal rank-form extension of Yau's conjecture

Let $M$ be a compact Kähler manifold with semi-negative holomorphic sectional curvature, and let $r^{-}_{\mathrm{tf}}$ denote the rank invariant defined in the paper. Let $\operatorname{kod}(M)$ be the Kodaira dimension, let $n(M)$ be the nef dimension, and let $K_M$ be the canonical line bundle.

The optimal rank-form extension of Yau's conjecture. One should have

$$\operatorname{kod}(M)=r^{-}_{\mathrm{tf}}.$$

In particular, if $r^{-}_{\mathrm{tf}}=\dim M$, then $K_M$ is ample. If $M$ is projective, one should moreover have

$$\operatorname{kod}(M)=r^{-}_{\mathrm{tf}}=n(M).$$

The source describes this as an optimal extension of Yau's conjecture. It is a theorem in the projective case in dimension at most $3$, and also under semi-negative bisectional curvature; the general compact Kähler and projective cases remain open.