Extend the Morse geodesic characterization beyond the affine-free setting

Extend the equivalence in Theorem 1.4 to all Coxeter groups by proving that, for any Coxeter group (without the affine-free assumption), a geodesic ray in the Davis complex is Morse if and only if (i) there exists a constant K such that every subpath longer than K has label not contained in any wide special subgraph, and equivalently (ii) the ray spends uniformly bounded time in cosets of wide special subgroups.

Background

Theorem 1.4 of the paper characterizes Morse geodesic rays in affine-free Coxeter groups via avoidance of wide special subgraphs and bounded time spent in cosets of wide special subgroups. This provides a practical criterion for Morse behavior but relies on the affine-free hypothesis, used through technical lemmas about wall intersections and recognition of affine subgroups.

The authors suggest that this characterization should hold more generally, and pose extending it to all Coxeter groups as a conjectural direction, which would remove the affine-free restriction and significantly broaden the applicability of their techniques.