The special 4-cycle center conjecture for Artin complexes

Let $A_S$ be an Artin group whose Coxeter diagram is a tree. A vertex of the Artin complex $\Delta_S$ is of type $\hat s=S\setminus\{s\}$ if it corresponds to a coset of the form $gA_{S\setminus\{s\}}$. A special $4$-cycle in $\Delta_S$ is an embedded $4$-cycle in the $1$-skeleton $\Delta_S^1$ whose vertex types alternate as $\hat s\hat t\hat s\hat t$ for some $s,t\in S$. Such a cycle has a center if its vertices are adjacent to a common vertex in $\Delta_S$. Special 4-cycle center conjecture. Any special $4$-cycle in $\Delta_S$ has a center.