Existence of purely axial torsion-wave solutions in quadratic metric-affine gravity

In quadratic metric-affine gravity, consider metric-compatible spacetimes with purely axial torsion waves and the field equations obtained by varying the action with respect to the metric and the connection. Purely axial torsion-wave conjecture. There exist purely axial torsion waves which are solutions of the field equations. This conjecture extends known purely tensor torsion-wave solutions to the purely axial case and asks whether Singh's Yang–Mills construction persists for the most general quadratic form with 16 coupling constants. The claim remains open in the source.