Karras' tame representation-type conjecture for normal surface singularities

Let $(A,\mathfrak{m})$ be a normal surface singularity over the algebraically closed field $k=A/\mathfrak{m}$ of characteristic zero. The tame representation-type conjecture. The ring $A$ is of tame Cohen–Macaulay representation type if and only if

$$A=B^G,$$

where $(B,\mathfrak{n})$ is either a simple elliptic or a cusp singularity and $G$ is a finite group of automorphisms of $B$. If $k=\mathbb{C}$, this is equivalent to the local fundamental group $\pi_1(X,o)$ being infinite and solvable. The conjecture aims to classify tame Cohen–Macaulay representation type among normal surface singularities; related results establish restrictions in the rational case, but the general classification remains open.