Eriksson's minimality criterion for canonical automata

Let $(W,S)$ be an irreducible Coxeter system. Let $\mathcal A_0(W,S)$ denote the canonical automaton recognizing the language of reduced words, let $\Sigma$ denote the set of small roots, and let $\Phi^+_{sph}$ be the set of roots whose support is a finite standard parabolic subgroup. Eriksson's minimality criterion. The automaton $\mathcal A_0(W,S)$ is minimal if and only if

$$\Sigma=\Phi^+_{sph}.$$

Minimality is known in type $\widetilde{A}_n$, while the canonical automaton is not minimal for general affine groups. The stated equivalence is presented as a conjectural necessary-and-sufficient condition, with the sufficient direction proved in the paper.