The spherical skeleton conjecture

Let $\mathscr{R}$ be a complete spherical skeleton, and let $P\subseteq G$ be the parabolic subgroup stabilizing the open $B$-orbit; the quantity $\dim G/P$ depends only on $\mathscr{R}$. Spherical skeleton conjecture. We have

$$\wp(\mathscr{R})\leq\dim G/P,$$

where equality holds if and only if $\mathscr{R}$ is linear, meaning that $\mathscr{R}\cong\mathscr{R}_V$ for some multiplicity-free space $V$. This is the general conjecture on spherical skeletons, with the supplied source giving no resolution status.