Finite degree formula for normalized volumes

Let

$$\pi\colon (y\in Y,D')\to (x\in X,D)$$

be a dominant finite morphism between klt singularities satisfying

$$K_Y+D'=\pi^*(K_X+D).$$

Finite degree formula. The normalized volumes satisfy

$$\deg(\pi)\cdot \widehat{\operatorname{vol}}(x,X,D)=\widehat{\operatorname{vol}}(y,Y,D').$$

The paper notes that this formula is known when $(X,x)$ lies on a Gromov–Hausdorff limit of Kähler–Einstein Fano manifolds, but presents the general statement as conjectural.