The divisorial valuation and rational-rank-one conjecture for symplectic singularities

Let $x\thicksim X$ be a symplectic singularity. Let $v_X$ be the valuation associated with $x$ in Theorem AGlc, and let $r(\xi)$ denote the rational rank associated with $\xi$ in Theorems DS.maps and AGlc. Divisorial valuation and rational-rank-one conjecture. For any symplectic singularity $x\in X$, the valuation $v_X$ is always an integer multiple of a divisorial valuation, and the rational rank $r(\xi)$ is always $1$. The conjecture is motivated by the preceding proposition and by the authors' examination of examples of symplectic singularities; its resolution is not specified in the supplied text.