The constant kth-mixed curvature conjecture for compact Hermitian manifolds

Let $(M^{n},g)$ be a compact Hermitian manifold with $n\geq2$. For $k\in\{1,2,3,4\}$ and $\beta\neq0$, define the kth-mixed curvature by

$$\mathcal{C}^{(k)}_{\alpha,\beta}(X)=\frac{\alpha}{|X|^{2}_{g}}Ric^{(k)}(X,\overline{X})+\beta H(X),$$

where $Ric^{(k)}$ is the kth Chern Ricci curvature and $H$ is the holomorphic sectional curvature. The constant kth-mixed curvature conjecture. If $\mathcal{C}^{(k)}_{\alpha,\beta}=c$ is constant and $c\neq0$, then $g$ is Kähler. The conjecture generalizes the mixed-curvature conjecture, which is known in complex dimension $2$ and for several special higher-dimensional classes; the general kth-mixed case remains open in the supplied text.