Chen–Zheng conjecture for Bismut connections

Let $(M,g)$ be a compact Hermitian manifold, and denote its Bismut connection by $\nabla^b$. The Bismut connection is the metric connection compatible with both the metric and the almost complex structure and having totally skew-symmetric torsion.

Chen–Zheng conjecture. If the holomorphic sectional curvature of $\nabla^b$ is a non-zero constant, then $g$ must be Kähler, hence a complex space form.

This extends the Chern and Levi-Civita constant-holomorphic-sectional-curvature conjecture to the Bismut connection. The source describes it as a conjecture raised in earlier work; the present paper verifies it for complex nilmanifolds with nilpotent complex structure and for non-balanced Bismut torsion-parallel manifolds, but the general case remains open.