The noncommutative arithmetic progression conjecture for compact quantum groups

Let $\mathbb{G}$ be the compact quantum group under consideration, let $\widehat{\mathbb{G}}$ be its set of irreducible representations, and let $\Sigma=\{F_n\}$ be a sequence of finite subsets of $\widehat{\mathbb{G}}$. For a subset $\Lambda\subseteq\widehat{\mathbb{G}}$, write $|\Lambda|_w$ for its weighted cardinality. Say that $\Lambda$ has positive upper density with respect to $\Sigma$ if

$$\limsup_{n\to\infty}\frac{|\Lambda\cap F_n|_w}{|F_n|_w}>0.$$

For $\alpha,\beta\in\widehat{\mathbb{G}}$, say that $\Lambda$ contains $\alpha^j\beta$ when every irreducible representation contained in the tensor-product representation $\alpha^j\beta$ belongs to $\Lambda$. Let $\widehat{\mathbb{G}}_\Sigma$ denote the subset of representations associated with the sequence $\Sigma$. Noncommutative arithmetic progression conjecture. If $\Lambda\subseteq\widehat{\mathbb{G}}$ has positive upper density with respect to $\Sigma$, then for every $k>0$ there exist $\alpha\in\widehat{\mathbb{G}}_\Sigma$, $\beta\in\widehat{\mathbb{G}}$, and $n>0$ such that $\Lambda$ contains $\alpha^{jn}\beta$ for every integer $j$ with $0\leq j\leq k-1$. This conjecture predicts arithmetic-progression-like configurations inside every positive-density subset of the dual object, extending the preceding arithmetic-progression theorem to the compact quantum-group setting. The supplied text does not state whether the conjecture is resolved.