Faithfulness conjecture for the Artin–Tits group action

Let $\Gamma$ be the underlying graph, let $\mathbb B$ be its Artin–Tits group, and let $\mathcal D$ be the associated triangulated category. The group $\mathbb B$ acts on $\mathcal D$ by autoequivalences.

Faithfulness conjecture. The action of the Artin–Tits group $\mathbb B$ on the triangulated category $\mathcal D$ is faithful.

By the main theorem, this is equivalent to the distinguished component $\operatorname{Stab}^\dagger_{\mathcal C}(\mathcal D)$ being simply connected. The paper explains that proving faithfulness would also solve the word problem in $\mathbb B$; the conjecture is not resolved here.