Non-necessity of the recurrence criterion in dimension two

Let $f_{e_i}$ denote the distance between consecutive points in coordinate direction $e_i$, and let $P$ and $\mathbb{E}$ denote the law and expectation used for the discrete point process. For $i\in\{1,2\}$ and $N\in\mathbb{N}$, consider the condition that there exists a constant $C>0$ such that

$$\sum_{k=N}^{\infty}\frac{k\,P(f_{e_i}=k)}{\mathbb{E}(f_{e_i})}\leq\frac{C}{N}.$$

Non-necessity conjecture. This condition, which is sufficient for recurrence of the two-dimensional random walk, is not necessary. The claim would show that recurrence can hold without the stated tail bound; the paper presents it as an open question in its partial classification of dimension two.