Neumann–Wahl conjecture for universal abelian covers

Let $(X,o)$ be a ${\mathbb Q}$-Gorenstein normal surface singularity with ${\mathbb Q}$-homology sphere link, and let $\Delta$ be the splice diagram associated to its resolution graph. An equisingular deformation here means a deformation of the splice-diagram complete intersection obtained by adding terms of the prescribed node weights while preserving the discriminant-group character action. Neumann–Wahl conjecture. The diagram $\Delta$ satisfies the semigroup condition, the resolution graph satisfies the congruence condition, and the universal abelian cover of $(X,o)$ is an equisingular deformation of the complete intersection of splice-diagram form. Moreover, the geometric genus of $(X,o)$ is determined by the topology. The paper proves the relevant conditions in several examples and the first nontrivial two-node cases, but the general realization and topological determination assertions remain open there.