Uniqueness conjecture for minimizing normalized-volume valuations

Let $(X,o)$ be a $\mathbb{Q}$-Gorenstein klt singularity, and let ${\rm Val}_{X,o}$ denote the space of valuations centered at $o$. The normalized volume is denoted by $\widehat{\rm vol}$.

Uniqueness conjecture. For any $\mathbb{Q}$-Gorenstein klt singularity $(X,o)$, there exists a unique valuation minimizing $\widehat{\rm vol}$ on ${\rm Val}_{X,o}$.

The existence of a minimizer is reduced in the paper to lower semicontinuity of the normalized volume, while uniqueness is presented as a conjectural strengthening. The claim concerns the local invariant obtained by minimizing normalized volume over valuations centered at the singular point.