Stringy Euler number criterion for smoothness of locally factorial spherical varieties

Let $X$ be a locally factorial spherical $G/H$-embedding whose closed orbits are projective. The stringy Euler number $e_{\rm st}(X)$ and the usual Euler number $e(X)$ are defined by $e_{\rm st}(X):=E_{\rm st}(X;1,1)$ and $e(X):=E(X;1,1)$.

Spherical smoothness conjecture. One has

$$e_{\rm st}(X)\geqslant e(X),$$

and equality holds if and only if $X$ is smooth.

This extends the corresponding theorem for locally factorial horospherical varieties to arbitrary locally factorial spherical varieties with projective closed orbits. The conjecture is motivated by Brion's smoothness criterion for spherical varieties, which is difficult to apply in practice.