The combinatorial generalized Mukai conjecture for spherical skeletons

Let $Rs\coloneqq(\Sigma,S^p,\mathcal{D}^a,\Gamma)$ be a complete spherical skeleton. Define

$$\mathcal{Q}^*_{\mathscr{R}}\coloneqq\bigcap_{D\in\Delta}\{v\in\Lambda_{\mathds{Q}}:\langle\rho(D),v\rangle\geq -m_D\}$$

and

$$\wp(\mathscr{R})\coloneqq\sup\left\{\sum_{D\in\Delta}\left(m_D-1+\langle\rho(D),\vartheta\rangle\right):\vartheta\in\mathcal{Q}^*_{\mathscr{R}}\cap\operatorname{cone}(\Sigma)\right\}\in\mathds{Q}_{\geq 0}\cup\{\infty\}.$$

Here $R$ is the underlying root system and $R_{S^p}$ is the root subsystem generated by $S^p$. Combinatorial generalized Mukai conjecture. We have

$$\wp(\mathscr{R})\leq |R^+\setminus R_{S^p}^+|.$$

Equality holds if and only if $Rs$ is isomorphic to the spherical skeleton $Rs_V$ of a multiplicity-free space $V$. This restates the variety-level conjecture combinatorially, and the paper uses it to study spherical skeletons and multiplicity-free spaces; no resolution status for this formulation is supplied in the candidate evidence.