Equivariant minimizer conjecture for klt singularities

Let $G$ be a group acting on a klt singularity $x\in (X,D)$, with $x$ fixed by the action. Let $\widehat{\operatorname{vol}}_{(X,D),x}$ denote the normalized-volume function. Group-action conjecture. There exists a $G$-invariant minimizer of the normalized volume. The conjecture is presented as a consequence of uniqueness in the stable degeneration conjecture; the torus case is known, while the finite-group case is related to the finite degree formula.