Finite-degree multiplication conjecture for local volumes

Let

$$\pi\colon x_1\in (X_1,\Delta_1)\to x_2\in (X_2,\Delta_2)$$

be a finite dominant morphism between klt singularities satisfying

$$\pi^*(K_{X_2}+\Delta_2)=K_{X_1}+\Delta_1.$$

Finite-degree multiplication conjecture. The local volumes satisfy

$$\deg(\pi)\cdot \operatorname{vol}(x_2,X_2)=\operatorname{vol}(x_1,X_1).$$

The source explains that this is related to uniqueness of the volume minimiser and records verification in a special class of Gromov–Hausdorff limits of Kähler–Einstein Fano manifolds; it remains a conjecture in general.