Generalized Yau conjecture for semi-negative holomorphic sectional curvature

Let $M$ be a projective manifold with a Kähler metric of semi-negative holomorphic sectional curvature. Define $r_M$ as the invariant measuring the largest codimension of a maximal subspace in a tangent space on which the holomorphic sectional curvature vanishes. The generalized Yau conjecture. The Kodaira dimension satisfies

$$\operatorname{kod}(M)\geq r_M.$$

In particular, if $r_M=\dim M$, then the canonical line bundle $K_M$ is ample. This generalizes the negative-curvature case and would relate the curvature condition to positivity of the canonical bundle; the source presents it as a conjectural strengthening, while proving partial results and the stated special-case consequences.