Ustinovskiy's weak Campana–Peternell conjecture

Let $(M,g)$ be a compact Hermitian manifold with Griffiths non-negative Chern curvature and positive first Chern--Ricci curvature. A rational homogeneous manifold is a quotient $\mathsf{P}\backslash\mathsf{G}$, where $\mathsf{G}$ is a complex semisimple Lie group and $\mathsf{P}\leq\mathsf{G}$ is a parabolic subgroup. Ustinovskiy's weak Campana–Peternell conjecture. Then $M$ is isomorphic to a rational homogeneous manifold $\mathsf{P}\backslash\mathsf{G}$. This conjecture proposes a uniformisation result for compact Hermitian manifolds under the stated positive curvature conditions; its resolution is not specified here.