Borbon–Spotti conjecture for local orbifold Euler numbers

Let $(X,D,x)$ be a germ of log canonical surface singularity with $\mathbb{Q}$-boundary. Borbon–Spotti conjecture.

$$e_{\rm orb}(x,X,D)= \begin{cases} \frac{1}{4}\widehat{\operatorname{vol}}(x,X,D), & \text{if }(X,D)\text{ is log terminal};\\ 0, & \text{if }(X,D)\text{ is not log terminal}. \end{cases}$$

The conjecture relates Langer's local orbifold Euler number to normalized volume and is motivated by volume densities of Kähler–Einstein metrics on surface singularities; it is known in special examples.