W*-rigidity conjecture for stationary stable random-field properties

Let $\mathcal{G}$ be a class of countable groups, let $\mathbb{F}_m$ denote the free group on $m$ generators, and let $W^\ast_m$-rigidity and $W^\ast_R$-rigidity refer to the rigidity properties defined in the paper. Free-group and property-(T) conjecture. Many probabilistic properties of stationary $S\alpha S$ random fields will become $W^\ast_m$-rigid for

$$\mathcal{G}=\{\mathbb{F}_m:m\in\mathbb{N}\}$$

as well as for

$$\mathcal{G}=\{G:G\text{ is a countably infinite group having property }(T)\},$$

and a few properties will in fact be $W^\ast_R$-rigid. The claim is deliberately broad; the source suggests beginning with particular properties such as ergodicity, weak mixing, and mixing, and gives no resolution.