Characterization of Morse geodesics without the affine-free hypothesis

Prove that for every Coxeter group, a geodesic ray in the Davis complex is Morse if and only if there exists a constant K such that any subpath of length greater than K does not have its label contained in a wide induced subgraph of the Coxeter graph, equivalently if and only if the ray spends uniformly bounded time in cosets of wide special subgroups.

Background

The paper proves this equivalence for affine-free Coxeter groups (Theorem \ref{thm:intro_thm_morse_char}), identifying a dynamical characterization of Morse geodesics via bounded time spent in wide special subgroups.

Extending this to all Coxeter groups would remove the affine-free restriction and yield a uniform criterion for Morse geodesics across the entire Coxeter class.