Recurrence of tangent-vector graphs for leaf-constructed measures

Let $T$ be a triangulation, let $x$ be a tangent vector on $T$, and let $G^x$ denote the corresponding graph of discrete tangent vectors. Suppose that $G^x$ is recurrent and that the measure $\nu_x$ exists. Recurrence conjecture. The graph $G^y$ is recurrent for $\nu_x$-almost every $y$. If true, this would imply amenability of $\hat{G}$ equipped with the transverse measure $\nu_x$ constructed under these conditions; the analogous amenability result is already proved for measures constructed as limits of measures on spheres.