The constant mixed curvature conjecture for compact Hermitian manifolds

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. Let $\alpha$ and $\beta$ be fixed real numbers with $\beta\neq0$, and let $\mathcal{C}_{\alpha,\beta}$ denote the mixed curvature. The constant mixed curvature conjecture. If

$$\mathcal{C}_{\alpha,\beta}=c$$

for a constant $c\neq0$, then $g$ is Kähler. For $n\geq3$, this conjecture is largely open; it is known under additional assumptions including Chern Kähler-like, locally conformally Kähler with $c\leq0$, and Bismut Kähler-like metrics, while the surface case is known.