Hypergraph-index conjecture for thickness and divergence of Coxeter groups

Let $(W,S)$ be a Coxeter system such that $W$ is $1$-ended, and let $h=h(W,S)$ be its hypergraph index. The hypergraph index measures the combinatorial complexity of the Coxeter system and gives an upper bound on both thickness and divergence.

Hypergraph-index conjecture. These upper bounds are equalities: the following are equivalent:

1. $(W,S)$ has finite hypergraph index $h$;
2. $W$ is strongly thick of order $h$;
3. the divergence of $W$ is polynomial of degree $h+1$.

The paper proves the corresponding upper bounds in general and establishes the conjecture for certain families of Coxeter groups. The converse in the quadratic case is known for right-angled Coxeter groups, while the general equivalence remains open.