The Riemannian constant holomorphic sectional curvature conjecture for compact Hermitian manifolds

Let $(M^n,g)$ be a compact Hermitian manifold of complex dimension $n\geq 2$, and let $H^r$ denote the Riemannian holomorphic sectional curvature, with $R^r$ the curvature of the Riemannian connection. Riemannian constant holomorphic sectional curvature conjecture. If

$$H^r=c$$

is constant, then $g$ must be Kähler when $c\neq 0$, and $g$ must be Riemannian flat, namely $R^r=0$, when $c=0$. This is the Riemannian twin of the Chern-connection conjecture. Its resolution status is not specified in the supplied text.