Non-hyperbolicity of projective manifolds with trivial real first Chern class

Let $X$ be a projective manifold, and let $c_1(X)$ denote its first Chern class in real cohomology.

Non-hyperbolicity conjecture. If

$$c_1(X)=0$$

in real cohomology, then $X$ is not Kobayashi hyperbolic.

The source introduces this statement as the reduction needed to derive Kobayashi's ampleness conjecture from the abundance conjecture. It is therefore presented as a conjectural step in the birational-geometric argument.