Critical-point shift criterion for pinning models with stable disorder

Let $\tau$ be the renewal process with inter-arrival law $K$, and let $h_c(\beta)$ denote the critical point at disorder intensity $\beta$. Assume that the environment satisfies the stable-tail assumption with parameter $\gamma$, and that $K$ is regularly varying. Critical-point shift criterion. The following equivalence holds:

$$\left\{\forall \beta>0,\ h_c(\beta)>0\right\}\quad\Leftrightarrow\quad\left\{\sum_{n\;\geqslant\; 1}\mathbf P[n\in\tau]^{\gamma}<\infty\right\}.$$

This would give a necessary and sufficient condition for disorder relevance at every temperature in the stable-environment pinning model. The paper indicates that one implication may be approachable by adapting known finite-variance techniques, whereas the converse is substantially more challenging.