Nonexistence of finite-dimensional local models for nonabelian quotient surface singularities

Let $G < \operatorname{GL}(2, \mathbb{C})$ be a nonabelian finite subgroup and let $R=\mathbb{C}\llbracket x,y\rrbracket^G$. A finite-dimensional local $\mathbb{C}$-algebra is an algebra that is finite-dimensional as a $\mathbb{C}$-vector space and has a unique maximal ideal. Nonexistence conjecture. There does not exist a finite-dimensional local $\mathbb{C}$-algebra $S$ such that

$$D_{sg}(R)\cong D_{sg}(S).$$

This asserts that the paper's Knörrer-type equivalence result is optimal for nonabelian quotient surface singularities: their singularity categories cannot be realized by finite-dimensional local algebras in the stated way. The source provides no resolution status beyond presenting this as a conjecture.