The Strominger–Bismut Kähler-like conjecture

Let $M$ be a compact complex manifold endowed with a Hermitian metric, and let its Strominger–Bismut connection be the Hermitian connection associated with that metric. The Strominger–Bismut Kähler-like conjecture. If the Strominger–Bismut connection is Kähler-like, then the metric is pluriclosed. This is motivated by the relationship between the Strominger–Bismut connection, Aeppli cohomology, and pluriclosed metrics. The conjecture has partial evidence from the fact that Strominger–Bismut-flat metrics are pluriclosed, while its general case remains open.