The fixed-point conjecture for elliptic subgroups of Artin groups

Let $A_\Gamma$ be an Artin group, let $C_\Gamma$ be its associated clique-cube complex, and let

$$\Phi\colon A_\Gamma\to {\rm Isom}(C_\Gamma)$$

be the action by left multiplication. For a subgroup $H\subset A_\Gamma$, write ${\rm Fix}(\Phi(H))$ for the common fixed-point set. The elliptic fixed-point conjecture. If $\Phi(h)$ is elliptic for every $h\in H$, then

$${\rm Fix}(\Phi(H))\ne\emptyset,$$

and consequently $\Phi(H)$ is contained in a complete parabolic subgroup. The result is known for finitely generated groups acting on finite-dimensional CAT$(0)$ cube complexes by elliptic isometries; this conjecture concerns arbitrary subgroups of Artin groups.