Equivariant smoothing of cusp singularities

We generalize Looijenga's conjecture for smoothing surface cusp singularities to the equivariant setting. Moreover, we prove that for any cusp singularity which admits a one-parameter smoothing, the smoothing can always be induced by smoothing of locally complete intersection cusps. The result provides evidence for the existence of the moduli stack of $\lci$ covers over semi-log-canonical surfaces.