Nilmanifold deformation conjecture for compact non-Kähler SKL manifolds

Let $(M^n,g)$ be a compact Strominger Kähler-like (SKL) manifold, meaning a Hermitian manifold equipped with a Strominger Kähler-like metric, with $g$ non-Kähler. Assume that its canonical line bundle is trivial. A complex nilmanifold is a quotient $N=G/\Gamma$, where $G$ is a nilpotent Lie group and $\Gamma$ is a cocompact lattice, equipped here with a left invariant metric compatible with a left invariant complex structure on $G$.

Nilmanifold deformation conjecture. Then $(M^n,g)$ can be deformed to a complex nilmanifold $(N^n,h)$ of this form. Moreover, the step of $G$ is at most two, the left invariant complex structure on $G$ is necessarily abelian, and its structure is the one described by Theorem 1 of ZZ1.

Complex nilmanifolds provide higher-dimensional analogues of Kodaira surfaces, and the cited classification suggests that they may account for compact non-Kähler SKL manifolds with trivial canonical bundle, up to deformation of complex structures and SKL metrics. The conjecture remains open in the supplied source.