Spectral selections, commutativity preservation and Coxeter-Lipschitz maps

For a Coxeter system $(W,S)$ a self-map $τ$ of $W$ such that $τ_{σθ}\in \{τ_θ,\ στ_θ\}$ for all $θ\in W$ and all reflections $σ$ (analogous to being 1-Lipschitz with respect to the Bruhat order on $W$) is either constant or a right translation. A somewhat stronger version holds for $S_n$, where it s