The gate set is a Garside shadow

Let $(W,S)$ be a Coxeter system, let $\Gamma$ be the set of gates of $W$, and let joins be taken in the weak order on $W$. A Garside shadow conjecture. The set $\Gamma$ is closed under join, and hence is a Garside shadow. This would resolve the conjecture of Hohlweg, Nadeau, and Williams that the smallest Garside shadow yields the minimal automaton. The paper proves this when $\mathcal{E}=\Phi_{\mathrm{sph}}^+$, but the general statement remains open.