Quasi-positive holomorphic sectional curvature and rational connectedness

Let $X$ be a compact Kähler manifold. Suppose it has a smooth Hermitian metric $\omega$ with quasi-positive holomorphic sectional curvature, or with uniformly RC-quasi-positive $(T_X,\omega)$.

Quasi-positive curvature conjecture. Then $X$ is projective and rationally connected.

The paper proves related results under stronger curvature hypotheses, including nonnegative holomorphic sectional curvature together with positivity away from a suitable subset, and everywhere positive holomorphic sectional curvature. The stated quasi-positive version is left as a conjecture.