Conjecture on quasi-positive holomorphic sectional curvature

Let $X$ be a compact Kähler manifold of dimension $n\geq 3$. A Kähler metric has quasi-positive holomorphic sectional curvature if its holomorphic sectional curvature is nonnegative everywhere and positive at some point.

Conjecture on quasi-positive curvature. If $X$ has a Kähler metric with quasi-positive holomorphic sectional curvature, then $X$ is a projective and rationally connected manifold.

The statement is known in complex dimension two from results cited in the paper, while the source proposes it for dimensions $n\geq 3$. It is presented as an open high-dimensional extension of Yau's conjecture.