The spectral characterization of geodesic growth for right-angled Coxeter groups on trees

Let $T$ and $T'$ be trees. Assume that they are simultaneously co-spectral, their complements are co-spectral, and their line graphs are co-spectral. Geodesic-growth conjecture. Then $T$ and $T'$ have the same geodesic growth, meaning that the associated right-angled Coxeter groups $G(T)$ and $G(T')$ have the same geodesic growth series. The conjecture asks whether these three spectral data determine geodesic growth for right-angled Coxeter groups based on trees; the paper establishes examples supporting the question but does not resolve the general implication.