Critical recurrence conjecture for stationary random walks on virtually--Z lamplighter groups

Let $G$ be a virtually-$\mathbb{Z}$ group and let $H$ be a nontrivial finite group. Let $\lambda_c$ be the critical parameter such that the $\lambda$-stationary random walk on $H\wr G$ is transient for $\lambda<\lambda_c$ and recurrent for $\lambda>\lambda_c$. Critical recurrence conjecture. The $\lambda$-stationary random walk on $H\wr G$ is recurrent also when $\lambda=\lambda_c$. This conjecture predicts that the phase transition established for virtually-$\mathbb{Z}$ base groups includes its critical parameter; the recurrence at $\lambda=\lambda_c$ remains open in the stated context.