ACC conjecture for local volumes

Let $d$ be a positive integer and $b$ be a set. For a klt singularity $(X\ni x,B)$, write $\widehat{\operatorname{vol}}(X\ni x,B)$ for its local volume. Consider the set

$$\operatorname{Vol}_{d,b}^{\operatorname{loc}}:=\left\{\widehat{\operatorname{vol}}(X\ni x,B)\Bigm| \dim X=d,B\inb,x\text{ is a closed point}\right\}.$$

ACC conjecture for local volumes. If $b$ is finite, then the only accumulation point of $\operatorname{Vol}_{d,b}^{\operatorname{loc}}$ is $0$. If $b$ satisfies the descending chain condition (DCC), then $\operatorname{Vol}_{d,b}^{\operatorname{loc}}$ satisfies the ascending chain condition (ACC). The paper states that the first assertion was proved by Xu and Zhuang and proves the second, so the conjecture is now solved.