Finite generation of the graded ring of a normalized-volume minimizer

Let $x uin (X=\operatorname{Spec} R,\Delta)$ be a germ of a klt singularity. Let $v$ be a quasi-monomial minimizer of $\widehat{\operatorname{vol}}_{X,\Delta}$.

Finite-generation conjecture. The associated graded ring $\operatorname{gr}_v R$ is finitely generated.

This is the remaining part of the Stable Degeneration Conjecture after the known results that minimizers are quasi-monomial and are log-canonical places of a $\mathbb{Q}$-complement. The statement is known in dimension $2$, but remains open in general.