The converse recurrence criterion for anisotropic random walks on the plane

Let $(p_j)_{j\in\mathbb Z}$ be the transition-probability parameters of the anisotropic walk, with $p_j>0$. The walk is called transient if it visits each state only finitely often almost surely. Converse recurrence criterion. If

$$\sum_{k=0}^{\infty}\left(\sum_{j=-k}^k \frac{1}{p_j}\right)^{-1}<\infty,$$

then the anisotropic walk is transient. This would establish the converse of the sufficient recurrence condition obtained earlier; whether the condition is also necessary for recurrence is posed as an intriguing open question.