Bergelson's amenability conjecture via recurrence

Let $G$ be a group. A set $R\subset G$ is a set of measurable recurrence if, for every measure-preserving $G$-system $(X,\mathcal{X},\mu,(T_g))$ and every $A\in\mathcal{X}$ with $\mu(A)>0$, there exists $g\in R^{*}$ such that

$$\mu(A\cap T_g^{-1}A)>0.$$

A set $R\subset G$ is a set of topological recurrence if it has the corresponding recurrence property for topological $G$-systems. The Bergelson amenability conjecture. The group $G$ is amenable if and only if every set of measurable recurrence $R\subset G$ is a set of topological recurrence.

This conjecture proposes a characterization of amenability through the equivalence of measurable and topological recurrence. The source notes that the implication from topological to measurable recurrence can fail in general, while amenable groups satisfy the forward implication; the converse characterization is presented as open.