Necessity of finite return-intersection sum for positive critical temperature

Let $\alpha\in(0,\infty)$, let $\chi:=\sum_{n\in\mathbb{N}}[\textbf{P}(S_n=0)]^2$, and let $\beta_c$ denote the critical inverse temperature separating the annealed and quenched critical curves. Return-intersection conjecture. If

$$\alpha\in(0,\infty)\quad\text{and}\quad\chi=\infty,$$

then $\beta_c=0$. This would show that the condition $\chi<\infty$ in the preceding sufficient condition for $\beta_c>0$ is also necessary; the conjecture was left for a forthcoming paper.