Critical recurrence and transience conjecture for biased random networks on lattices

Let $p_c$ denote the critical probability for Bernoulli bond percolation on ${\mathbb{Z}}^d$, and let $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p)$ denote the biased disordered random network with conductance parameters $\lambda_1$ and $\lambda_2$. Critical recurrence and transience conjecture. When $3\leq d\leq 10$ and $0<\lambda_1\leq 1<\lambda_2$, almost surely $({\mathbb{Z}}^d,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,p_c)$ is recurrent. Moreover, the biased network $({\mathbb{Z}}^2,\, \mathbf{C}_{\lambda_1},\, \mathbf{C}_{\lambda_2},\,1/2)$ with $0<\lambda_1<\lambda_2=1$ is transient almost surely. This extends the known critical-percolation picture, where non-existence of an infinite cluster at criticality is known in dimensions $2$ and at least $11$, while the corresponding question remains open in dimensions $3$ through $10$; the competing biased behavior in dimension two is expected to be transient.