Equivalence of hypergraph index, thickness and divergence for right-angled Coxeter groups

Let $\Gamma$ be a simplicial graph and let $W_{\Gamma}$ be the corresponding right-angled Coxeter group. Fix an order $n$. Equivalence conjecture. The following are equivalent: $\Gamma$ has hypergraph index $n$; $W_{\Gamma}$ is thick of order $n$; and the divergence of $W_{\Gamma}$ is a polynomial of degree $n+1$. For $n=0$ and $n=1$, these notions are known to be equivalent within the class of right-angled Coxeter groups; the conjecture proposes the equivalence in general, and is presented as potentially extending to groups whose divergence and thickness can be computed.