Strictness conjectures for hyperbolic structures on spherical Artin–Tits groups

Let $A=A_S$ be an irreducible Artin–Tits group of spherical type with rank at least $3$. Let $X_{abs}^A$ be the absorbable-element generating set, $X_{NP}^A$ the union of normalizers of proper irreducible standard parabolic subgroups, and $X_P^A$ the union of proper irreducible standard parabolic subgroups together with the cyclic subgroup generated by $\Delta_S^2$. Write $X\preccurlyeq Y$ when the identity map from $(A,d_Y)$ to $(A,d_X)$ is Lipschitz. Strictness conjectures. (i) The inequality $X_{abs}^A\preccurlyeq X_{NP}^A$ is not strict: $X_{NP}^A\preccurlyeq X_{abs}^A$ also holds, and the identity map $\Gamma(A,X_{NP}^A)\to\Gamma(A,X_{abs}^A)$ is a quasi-isometry. (ii) The inequality $X_{NP}^A\preccurlyeq X_P^A$ is strict: the identity map $\Gamma(A,X_P^A)\to\Gamma(A,X_{NP}^A)$ is Lipschitz but not a quasi-isometry. These claims compare the large-scale geometry of three hyperbolic structures; the source presents them as open problems, with the braid-group instance in (i) relating the additional length graph to the curve graph of the punctured disk.