The growth–random-walk equivalence for finitely generated quantum groups

Let $(A,u)$ be a finitely generated quantum group, and let $b_k$ and $p_k$ denote the growth and random-walk quantities constructed in the paper. Here $f_k\approx g_k$ means that the two sequences have the comparison relation used in the paper. Growth–random-walk conjecture. One has

$$b_k\approx k^d \quad\text{if and only if}\quad p_k\approx k^{-d/2}.$$

This extends to discrete quantum groups estimates known for discrete groups, where the equivalence is a key ingredient in Gromov's fundamental result. The source does not provide a resolution, so the statement remains open.