Kobayashi's ampleness conjecture for compact Kobayashi-hyperbolic manifolds

Let $X$ be a compact Kähler or projective manifold that is Kobayashi hyperbolic.

Kobayashi's conjecture. The canonical bundle $K_X$ is ample.

This conjecture asks whether Kobayashi hyperbolicity forces positivity of the canonical bundle. The paper notes that it has a positive answer for compact Kähler hyperbolic manifolds, but the general case remains open.