Poisson boundary conjecture for bounded-range lamplighter walks on oriented trees

Let $\mathcal{T}$ be an oriented tree with a fixed end, and let $(Z_n)$ be a random walk with bounded range on $\mathbb{Z}_2\wr\mathcal{T}$. Write $(X_n)$ for its projection to $\mathcal{T}$, and suppose that $(X_n)$ has zero modular drift. The lamp configuration is said to have an infinite limit configuration when it converges pointwise along the walk. Poisson boundary conjecture. The Poisson boundary of $(Z_n)$ is isomorphic to the space of infinite limit configurations of lamps, endowed with the respective hitting distribution. This would extend the nearest-neighbour result proved in the paper to bounded-range walks and bounded-range lamp changes; the claim remains open because the proof established only the nearest-neighbour case.