Stevens's tautness and quasi-homogeneity conjecture for simple surface singularities

Let $X$ be a normal surface singularity. It is taut if every other singularity with the same resolution graph is isomorphic to $X$, and quasi-homogeneous if it admits a compatible weighted grading. It is simple if only finitely many isomorphism classes occur in its versal deformation.

Stevens's conjecture. A normal surface singularity is simple if and only if it is taut and quasi-homogeneous.

This formulation is presented as equivalent to the graph-theoretic characterization in the paper, using Laufer's classification of taut and pseudotaut singularities. The source does not provide a resolution status for this formulation.