The amenability characterization of richly synergodic groups

Let $G$ be a countably infinite group. Recall that $G$ is richly synergodic if the synergodic actions are dense in $\mathrm{Erg}(G \times G,X,\mu)$ for every standard probability space $(X,\mu)$. Amenability conjecture for rich synergodicity. The group $G$ is richly synergodic if and only if it is amenable.

This conjecture would characterize rich synergodicity in purely group-theoretic terms. Every countably infinite amenable group is richly synergodic, and groups containing a nonabelian free subgroup are known not to be richly synergodic; the case of nonamenable groups without free subgroups remains open.