The generic smoothness and Euler-number conjecture for the McKay polyhedra

Let $C_\zeta$ be the polyhedra associated with the parameter $\zeta$, let $n$ denote the ambient dimension, and let $\mathcal{S}^k$ be the configurations of codimension $k$ whose associated tangent cones are singular. Let $\mathcal{S}_{\operatorname{span}}^0$ denote the configurations of codimension zero occurring for generic parameters, and let $\mathcal{T}_\zeta^0/\!\!\sim$ denote the corresponding equivalence classes of zero-dimensional trees. Generic smoothness and Euler-number conjecture. The polyhedra $C_\zeta$ are nonsingular in codimension $n-1$, equivalently $\mathcal{S}^1=\varnothing$. If $\Gamma\subset\operatorname{SU}(3)$, then $C_\zeta$ is nonsingular for generic values of $\zeta$, equivalently $\mathcal{S}_{\operatorname{span}}^0=\varnothing$. Furthermore, for generic $\zeta$ its Euler number is the order of $\Gamma$:

$$\sharp\,\mathcal{T}_\zeta^0/\!\!\sim=\sharp\,\Gamma.$$

These assertions record the expected smoothness properties of the slices and the count of their zero-dimensional strata; the source gives no evidence that they have been proved or disproved.