Ergodicity of the uniform transverse measure

Let $\nu_u$ be the limiting transverse measure obtained from a convergent sequence of uniform probability measures on finite sphere triangulations, and let $G$ be the associated groupoid equipped with this measure. Ergodicity conjecture. The measure $\nu_u$ is ergodic. The paper proves ergodicity for measures whose rooted patch densities are almost surely independent of the root, but the supplied text does not establish that $\nu_u$ satisfies this hypothesis, so the claim remains open here.