Supersmoothing conjecture for pinning models with stable disorder

Let $F(\beta,h)$ be the quenched free energy, let $h_c(\beta)$ be the critical point, and suppose the renewal inter-arrival law satisfies the regular-variation assumption and the environment satisfies the stable-tail assumption with parameter $\gamma>1$. Supersmoothing conjecture. For every $\beta>0$, there exists a constant $C_\beta>0$ such that, for all $u\in[0,1]$,

$$F(\beta,h_c(\beta)+u)\;\leqslant\; C_\beta u^{\frac{\gamma}{\gamma-1}}.$$

The conjecture predicts a free-energy curve smoother than the quadratic bound known for general disordered pinning models. The paper motivates it through localization by large environmental fluctuations, but notes that turning this heuristic into a proof presents serious technical obstacles.