The periodic-order bound conjecture for finite orbits in spherical Artin–Tits groups

Let $G$ be an Artin-Tits group of spherical type, let $\mathcal C(G)$ be its curve graph, and let $\alpha\in G$. Suppose a vertex of $\mathcal C(G)$ has a finite orbit under $\alpha$. Periodic-order bound conjecture. The bound on the size of this finite orbit predicted by the finite-orbit bound conjecture is the maximal order of a periodic element in $G$. The source presents this as a refinement of the finite-orbit bound conjecture; it proves the preceding bound for braid groups but gives no resolution of this sharper assertion for general spherical Artin–Tits groups.