The Bismut holonomy existence conjecture for compact complex manifolds

Let $M$ be a compact complex manifold of real dimension $2n$ with vanishing first Chern class. A hermitian structure on $M$ is a Riemannian metric compatible with its complex structure; its Bismut connection is the hermitian connection whose torsion is a three-form. The Bismut holonomy conjecture. There exists a hermitian structure on $M$ whose Bismut connection has restricted holonomy contained in $SU(n)$:

$$\operatorname{Hol}_0(\nabla^{\mathrm{Bismut}})\subseteq SU(n).$$

This is proposed as a non-Kähler analogue of the Calabi conjecture, whose Kähler case is solved by Yau; the source does not provide a resolution for the non-Kähler assertion.