Yang–Zheng's ampleness conjecture for Hermitian manifolds

Let $X$ be a compact complex manifold, let $K_X$ denote its canonical bundle, and let holomorphic sectional curvature refer to the curvature of a Hermitian metric on $X$. A Hermitian metric has quasi-negative holomorphic sectional curvature if its holomorphic sectional curvature is nonpositive everywhere and negative at some point, and has negative holomorphic sectional curvature if it is negative everywhere. Yang–Zheng's conjecture.

$$\begin{array}{ll} (1) & \text{if } X \text{ is Kobayashi hyperbolic, then } K_X \text{ is ample};\\ (2) & \text{if } X \text{ admits a Hermitian metric with quasi-negative holomorphic sectional curvature, then } K_X \text{ is ample};\\ (3) & \text{if } X \text{ admits a Hermitian metric with negative holomorphic sectional curvature, then } K_X \text{ is ample}. \end{array}$$

These assertions concern the largely open relationship between curvature of arbitrary Hermitian metrics and positivity of the canonical bundle. In the Kähler setting, the corresponding ampleness results are known, while the paper establishes further partial results for Hermitian manifolds, so the conjecture remains open.