Compact K\"ahler manifolds with quasi-positive second Chern-Ricci curvature

Let $X$ be a compact K"ahler manifold. We prove that if $X$ admits a smooth Hermitian metric $\omega$ with quasi-positive second Chern-Ricci curvature ${\mathrm{R}\mathrm{i}\mathrm{c}}^{(2)}(\omega )$, then $X$ is projective and rationally connected. In particular, $X$ is simply connected.