Kobayashi–Lang conjecture on ampleness of the canonical bundle

Let $X$ be a compact complex manifold that is Kobayashi hyperbolic, meaning that its intrinsic Kobayashi pseudometric is a metric. Let $K_X$ denote its canonical line bundle.

Kobayashi–Lang conjecture. The canonical line bundle $K_X$ is ample.

This conjecture predicts a strong birational-geometric consequence of Kobayashi hyperbolicity. The paper proves ampleness under the stronger assumption of nondegenerate negative total $k$-jet curvature, but the conjecture itself is not resolved here.