Hyperbolicity of the parabolic generating set for spherical Artin–Tits groups

Let $A=A_S$ be a non-cyclic irreducible Artin–Tits group of spherical type. Let $X_P^A$ be the union of all proper irreducible standard parabolic subgroups of $A$ and the cyclic subgroup generated by the square of the Garside element $\Delta_S$. Hyperbolicity conjecture. The set $X_P^A$ is a hyperbolic structure on $A$. This is proved in the source for dihedral groups and groups of type $A_n$ with $n\geqslant 3$, while the assertion for arbitrary non-cyclic irreducible spherical type remains open there.