Spectral selections, commutativity preservation and Coxeter-Lipschitz maps

Let $(W,S)$ be a Coxeter system whose graph is connected, with no infinite edges. A self-map τ of W such that τσθ∈ \τθ,\ στθ\ for all θ∈ 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 Sₙ, where it suffices that σ range over smaller, θ-dependent sets of reflections. These combinatorial results have a number of consequences concerning continuous spectrum- and commutativity-preserving maps SU(n)→ Mₙ defined on special unitary groups: every such map is a conjugation composed with (a) the identity; (b) transposition, or (c) a continuous diagonal spectrum selection. This parallels and recovers Petek's analogous statement for self-maps of the space Hₙ≤ Mₙ of self-adjoint matrices, strengthening it slightly by expanding the codomain to Mₙ.