RC-positivity conjecture for exterior powers under positive holomorphic sectional curvature

Let $X$ be a compact complex manifold of complex dimension $n>2$. RC-positivity conjecture. If $X$ has a Hermitian metric with positive holomorphic sectional curvature, then $\Lambda^pT_X$ admits a smooth RC-positive metric for every $1<p<n$. In particular, if $X$ is Kähler, then $X$ is a projective and rationally connected manifold. The source explicitly proposes this statement; its resolution status is not given.