Universal critical-curve asymptotics for disordered pinning models

Let $h_c(\beta)$ be the critical curve of a disordered pinning model, and let $\boldsymbol h_c^{(\alpha)}(1)$ be the critical curve of the continuum model at unit coupling. For a slowly varying function $L$, define $\widetilde L$ through

$$\widetilde L(x)=\left[((\bar L_{\alpha-\frac12})^*)(x)\right]^{-\frac{1}{2\alpha-1}},$$

where $\bar L_{\alpha-\frac12}(x)=1/L(x^{1/(\alpha-1/2)})$ and $(\bar L_{\alpha-1/2})^*$ is its asymptotic inverse in the sense specified in the source. Universal pinning critical-curve conjecture. For any disordered pinning model satisfying with $\alpha\in(\frac12,1)$,

$$\lim_{\beta\downarrow0}\frac{h_c(\beta)}{\widetilde L(\frac1\beta)\beta^{\frac{2\alpha}{2\alpha-1}}}=\boldsymbol h_c^{(\alpha)}(1).$$

This predicts that the weak-disorder critical curve has a universal power-law exponent and a slowly varying correction determined by the renewal tail. The supplied text gives no resolution.