The negative second scalar curvature conjecture for rationally connected Kähler manifolds

Let $M^m$ be a compact Kähler manifold with $S_2<0$ and $m\geq 3$, where $S_2$ denotes the second scalar curvature, and let rationally connected mean that $M$ is rationally connected.

Negative second scalar curvature conjecture. The manifold $M$ cannot be rationally connected.

This is presented as a wild guess motivated by the scarcity of examples. The source gives no resolution of the claim, while posing related questions about negative scalar curvature on rationally connected manifolds.