The unique minimal parabolic subgroup conjecture for Artin groups

Let $B$ be an Artin group, and let ${\mathbf b}\in B$. A minimal parabolic subgroup containing ${\mathbf b}$ is a parabolic subgroup containing ${\mathbf b}$ that is minimal under inclusion. Unique minimal parabolic subgroup conjecture. There exists a unique minimal parabolic subgroup containing every given ${\mathbf b}\in B$. The conjecture follows if intersections of parabolic subgroups are parabolic, and the source notes supporting results for several families of Artin groups; it remains open in general.