The generalized degree-of-satisfiability conjecture for reasonable measures

Let $G$ be a group, let $\{\mu_n\}_{n\in \mathbb{N}}$ be a sequence of “reasonable” measures on $G$, and let $\mathcal{E}\subseteq F_k$ be a set of equations in $k$ variables. The degree of satisfiability is

$$\operatorname{ds}(G,\mathcal{E},\{\mu_n\}_{n\in \mathbb{N}})=\limsup_{n\to\infty}\mu_n^{\times k}\big(\{(g_1,\ldots,g_k)\in G^k:(g_1,\ldots,g_k)\text{ is a solution of }\mathcal{E}\}\big).$$

For the commutativity equation, write $\operatorname{dc}(G,\{\mu_n\}_{n\in\mathbb{N}})$ for this degree. Generalized degree-of-satisfiability conjecture. Then

$$\operatorname{ds}(G,\mathcal{E},\{\mu_n\}_{n\in\mathbb{N}})>0\quad\Longleftrightarrow\quad\mathcal{E}\text{ is a virtual law in }G.$$

In particular,

$$\operatorname{dc}(G,\{\mu_n\}_{n\in\mathbb{N}})>0\quad\Longleftrightarrow\quad G\text{ is virtually abelian}.$$

The conjecture proposes that positive asymptotic satisfaction probability is equivalent to an algebraic finite-index law, for suitable measure sequences. The meaning of “reasonable” measures is left informal in the source, so the precise scope and validity of the assertion remain open.