The admissible-cycle filling complexity conjecture for spherical Artin complexes

Let $A_S$ be an irreducible spherical Artin group, let $\Delta_S$ be its associated Artin complex, and let $\mathsf C_S$ be its associated Coxeter complex with the natural action of $W_S$. A wall of $\mathsf C_S$ is the fixed-point set of a reflection of $W_S$; its complement has two connected components, called open halfspaces. An $n$-cycle is admissible if every $n$-cycle in $\mathsf C_S$ with the same type sequence is contained in some open halfspace. For any admissible $n$-cycle $\omega$ of $\Delta_S$, there is a companion $n$-cycle $\omega'$ in $\mathsf C_S$ with the same type sequence as $\omega$ such that the minimal disk filling $\omega$ in the 2-skeleton of $\Delta_S$ is no more complicated than the minimal disk filling $\omega'$ in the 2-skeleton of $\mathsf C_S$. The conjecture proposes that Coxeter-complex fillings control the combinatorial complexity of fillings in spherical Artin complexes; it is motivated by earlier results and is intended to guide the study of the $K(\pi,1)$ conjecture, but its general status is unresolved in the source.