The rational connectedness conjecture for compact Kähler manifolds

Let $X$ be a connected compact Kähler manifold. For integers $k>0$ and $p>0$, write $S^k\Omega_X^p$ for the $k$th symmetric power of the sheaf of holomorphic $p$-forms.

Rational connectedness conjecture. $X$ is rationally connected, meaning that any two generic points are joined by some rational curve, if and only if

$$H^0(X,S^k\Omega_X^p)=0$$

for every $k>0$ and $p>0$.

A weaker version, usually attributed to D. Mumford, asks for the same conclusion under the assumption that $H^0(X,(\Omega_X^1)^{\otimes k})=\{0\}$ for all $k>0$. The conjecture proposes a criterion for rational connectedness through the vanishing of all symmetric pluridifferentials.