Antolín–Martino–Ventura conjecture on degree of commutativity in exponential-growth groups

Let $G$ be a finitely generated group of exponential word growth, and let $S$ be a finite generating set of $G$. Antolín–Martino–Ventura conjecture. Then

$$\operatorname{dc}_S(G)=0,$$

irrespective of the choice of $S$.

For groups of sub-exponential word growth, positivity of the degree of commutativity is equivalent to virtual abelianness, and the degree is independent of the finite generating set. The conjecture asks whether every finitely generated group of exponential word growth has degree of commutativity zero for every finite generating set; its status is not determined by the supplied source.