Zheng's hyperbolicity conjecture for nonpositively curved Kähler manifolds

Let $M$ be a nonpositively curved compact Kähler manifold. Zheng's conjecture. If $M$ is of general type, then it must be Kobayashi hyperbolic, meaning that every holomorphic map

$$f:\mathbb{C}\rightarrow M$$

is constant. The conjecture is attributed to F. Zheng. The source gives a related theorem establishing ampleness of the canonical bundle under stronger hypotheses, but does not state that this conjecture has been resolved.