Newton non-degeneracy characterization for normal surface singularities

A normal surface singularity is a surface singularity that is normal, and a Newton non-degenerate singularity is one for which, for every $\mathbf{u}\in\mathbb{R}_{\geq 0}^n$, the variety defined by its initial ideal has no singularity in $(\mathbb{C}^*)^n$. A normalisation is the normal variety obtained by normalizing a singularity, and a nonisolated non-degenerate hypersurface singularity is a nonisolated hypersurface singularity satisfying the stated Newton non-degeneracy condition. Newton non-degeneracy characterization. A normal surface singularity is Newton non-degenerate if and only if it is a normalisation of a nonisolated non-degenerate hypersurface singularity. The theorem preceding this conjecture establishes Newton non-degeneracy for the nonisolated forms of rational triple point singularities and their normalisations, but the claimed characterization for all normal surface singularities remains open.