Parabolic intersection conjecture for Artin–Tits groups

Let $A_I$ be an Artin–Tits group. For $X\subseteq I$, the standard parabolic subgroup $A_X$ is the subgroup generated by the standard generators indexed by $X$; a parabolic subgroup is a conjugate of a standard parabolic subgroup. Let $X,Y\subseteq I$ and let $\boldsymbol{\omega}\in A_I$. Parabolic intersection conjecture. The subgroup

$$(\boldsymbol{\omega}A_Y\boldsymbol{\omega}^{-1})\cap A_X$$

is a parabolic subgroup of $A_I$. This is a main open problem for Artin–Tits groups. It is known for spherical, FC and large type Artin–Tits groups, but remains open in general.