Cheliotis–den Hollander critical-temperature conjecture for pinning models

Consider a renewal pinning model with parameter $a\in(0,\infty)$, disorder law $\mu_0$, and critical inverse temperature $\beta_c$. Let

$$\chi=\sum_{n\in\mathbb N}[P(S_n=\ast)]^2$$

denote the expected number of meetings at $\ast$ of two independent copies of the Markov chain.

Cheliotis–den Hollander conjecture. For every $\mu_0$,

$$\chi=\infty \quad\Longrightarrow\quad \beta_c=0.$$

This is described as a challenging problem and is proved in the source only under more restrictive assumptions on the return law $R$; therefore the general assertion remains open.