Chen–Nie conjecture for canonical Hermitian connections

Let $D^r_s$ be a canonical Hermitian connection indexed by $(r,s)\in\Omega$, let $\Gamma\subset\Omega$ be the Chen–Nie curve

$$\Gamma=\{(r,s)\in\mathbb{R}^2\mid (1-r+rs)^2+s^2=4\},$$

and let $D^r_s$-flat mean that the curvature of $D^r_s$ vanishes.

Chen–Nie conjecture. Let $(M^n,g)$ be a compact Hermitian manifold. Assume that the holomorphic section curvature of $D^r_s$ is a constant $c$. If $c\neq 0$, then $g$ must be Kähler. If $c=0$ and $(r,s)\in\Omega\setminus\Gamma$, then $g$ must be $D^r_s$-flat.

The conjecture is known in complex dimension two and for several special classes, including complex nilmanifolds with nilpotent complex structure and certain Bismut torsion-parallel manifolds. The general higher-dimensional problem, especially on the Chen–Nie curve, remains open.