Arbitrary intersection of parabolic subgroups in general Artin groups

Determine whether, for an arbitrary Artin group A_S defined by generators S with standard Artin relations, the intersection of an arbitrary family of parabolic subgroups—i.e., subgroups conjugate to standard parabolic subgroups generated by subsets of S—is itself a parabolic subgroup.

Background

Artin groups A_S are defined by generators S and relations of the form sts⋯=tst⋯ determined by a Coxeter matrix. Parabolic subgroups are conjugates of standard parabolic subgroups generated by subsets of S and play a central role in the paper of these groups and associated complexes. While the paper provides a concise proof that the arbitrary intersection of parabolic subgroups in the Euclidean braid groups A[~A_n] is parabolic, the broader question remains unresolved for most other Artin groups.

It is known that in spherical-type Artin groups, the intersection of parabolic subgroups is parabolic. The authors highlight that, beyond these cases (and the Euclidean case addressed here), the general property is not established, underscoring a significant open direction in the theory of Artin groups.