Ergodicity conjecture for the Furstenberg–Poisson-boundary stable random field

Let $\mathbb{F}_m$ act on its Furstenberg–Poisson boundary $\partial\mathbb{F}_m$, and let the stationary $S\alpha S$ random field associated with this boundary action be the one considered in Example3.2 of Sarkar and Roy. Boundary-action ergodicity conjecture. The stationary $S\alpha S$ random field considered in Example3.2 of Sarkar and Roy, generated by the boundary action of $\mathbb{F}_m$, is ergodic. The conjecture is presented as a concrete starting point for the broader rigidity conjecture; the source gives no resolution.