The polynomial growth conjecture for discrete quantum groups

Let $G\subset U_N^+$ be a compact quantum group, and let $\widehat{G}$ denote its discrete dual. For each $Q\in U_N$, let $\Gamma_Q$ be the associated discrete quantum group. Polynomial growth is defined using the balls in $\operatorname{Irr}(G)$ for the distance induced by the fundamental corepresentation, with each corepresentation weighted by the square of its dimension. Growth conjecture. The discrete quantum group $\widehat{G}$ has polynomial growth if and only if every $\Gamma_Q$ has polynomial growth.

This conjecture proposes that polynomial growth is detected exactly by the family of quantum group fibers $\Gamma_Q$. The source gives the formulation but no general resolution, so its status remains open.