Antolín, Martino and Ventura's meta-conjecture on degree of commutativity

Let $G$ be a group, and let $M=(\mu_n)_{n=1}^\infty$ be a sequence of probability measures on $G$. For a probability measure $\mu$ on $G$, define

$$\operatorname{dc}_\mu(G)=(\mu\times\mu)(\{(x,y)\in G\times G:xy=yx\}),$$

and define

$$\operatorname{dc}_M(G)=\limsup_{n\to\infty}\operatorname{dc}_{\mu_n}(G).$$

Antolín, Martino and Ventura's meta-conjecture. For any “reasonable” sequence $M=(\mu_n)_{n=1}^\infty$ of probability measures on $G$, we have $\operatorname{dc}_M(G)>0$ if and only if $G$ is virtually abelian.

This conjecture proposes a natural infinite-group analogue of the positive degree-of-commutativity result for finite groups. Its formulation depends on the meaning of “reasonable” for the sequence of measures; the source presents it as a meta-conjecture, and no resolution is supplied here.