Yau's projectivity and rational connectedness conjecture for positively curved Kähler manifolds

Let $M$ be a compact complex manifold admitting a Kähler metric $\omega$ with positive holomorphic sectional curvature. The holomorphic sectional curvature along a nonzero $V\in T^{1,0}M$ is

$$H(V)=\frac{R(V,\overline{V},V,\overline{V})}{\lvert V\rvert_{\omega}^4},$$

where $R$ is the Riemannian curvature tensor of $\omega$. Yau's conjecture. If $M$ admits such a metric, then $M$ is projective and rationally connected. This conjecture gives a geometric characterization of rational connectedness in terms of positive holomorphic sectional curvature. The supplied text does not state whether it has been resolved.