Yang's Hermitian generalization for exterior powers of the tangent bundle

Let $X$ be a compact complex manifold of complex dimension $n>2$, and suppose that $X$ has a Hermitian metric with positive holomorphic sectional curvature. A Hermitian vector bundle is RC-positive if it admits a smooth metric with the RC-positivity property.

Yang's conjecture. For every integer $p$ with $1<p<n$, the bundle $\bigwedge^p T_X$ admits a smooth RC-positive metric. In particular, if $X$ is Kähler, then $X$ is a projective and rationally connected manifold.

The paper's theorem establishes vanishing results and partially confirms this conjecture under the stronger assumption of positive real bisectional curvature. The conjecture remains unresolved in the stated general Hermitian setting.