RC-positivity conjecture for maps to Kobayashi hyperbolic manifolds

Let $M$ and $N$ be compact complex manifolds. The line bundle $\mathcal O_{T^*M}(-1)$ is assumed to be RC-positive, and $N$ is assumed to be Kobayashi hyperbolic.

RC-positivity conjecture. There is no non-constant holomorphic map from $M$ to $N$.

This conjecture seeks a differential-geometric obstruction to holomorphic maps from manifolds with the stated RC-positivity property to Kobayashi hyperbolic targets. The supplied source does not indicate that it has been resolved and notes related versions elsewhere.