Every complete non-Kähler SKL threefold has universal cover either a product of two Sasakian 3-manifolds or a non-Kähler SKL surface times a Kähler curve, and degenerate-torsion SKL manifolds split off a Kähler factor.
Bismut, A local index theorem for non-K¨ ahler manifolds, Math
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On Strominger K\"ahler-like manifolds with degenerate torsion
Every complete non-Kähler SKL threefold has universal cover either a product of two Sasakian 3-manifolds or a non-Kähler SKL surface times a Kähler curve, and degenerate-torsion SKL manifolds split off a Kähler factor.