The Bismut Alekseevski–Kimelfeld conjecture

Let $(M^n,g)$ be a compact (or complete) Hermitian manifold. Its Bismut connection is denoted by $\nabla^b$, and a connection is Ambrose–Singer when its torsion and curvature are parallel with respect to itself. The first and third Bismut Ricci curvatures are the corresponding contractions of the Bismut curvature. Bismut Alekseevski–Kimelfeld conjecture. If $\nabla^b$ is Ambrose–Singer and the first and third Bismut Ricci curvatures both vanish, then the Bismut curvature vanishes. This is the proposed Bismut-connection analogue of the Alekseevski–Kimelfeld theorem. The statement is known in complex dimension $2$ in the compact case, while the authors suspect a negative answer to the converse question in dimensions at least $3$ but do not have a concrete counterexample; the conjecture itself is presented as open.