Gagliardi–Hofscheier spherical-skeleton conjecture

Let $\mathcal R=(\Delta,S^p,\Sigma,\Gamma)$ be a complete spherical skeleton, where $R$ is the underlying root system, $R^+$ is its set of positive roots, and $R^+_{S^p}$ is the set of positive roots in the linear span of $S^p$. Let $\mathcal R_V$ denote the spherical skeleton of a spherical $G$-module $V$. Gagliardi–Hofscheier spherical-skeleton conjecture. One has

$$\mathcal P(\mathcal R)\leq |R^+|-|R^+_{S^p}|,$$

and equality holds if and only if $\mathcal R$ is isomorphic to $\mathcal R_V$ for a spherical module $V$. This is a reformulation of the Gagliardi–Hofscheier conjecture for complete spherical varieties; the paper proves the corresponding result in its stated class, while the general spherical-skeleton formulation is not presented as settled.