Minimality conjecture for the smallest Garside-shadow automaton

Let $(W,S)$ be a Coxeter system, let $\tilde S$ be its smallest Garside shadow, and let $\mathcal A_{\tilde S}(W,S)$ be the associated finite deterministic automaton. Write $\operatorname{Red}(W,S)$ for the language of reduced words of $(W,S)$. Minimality conjecture. The automaton $\mathcal A_{\tilde S}(W,S)$ is the minimal automaton recognizing $\operatorname{Red}(W,S)$. The main theorem shows that this automaton recognizes the language; the conjecture concerns whether it has the smallest possible number of states. The paper gives surjective morphisms from automata associated with larger Garside shadows, but does not establish minimality.