Infinite-order criticality conjecture for the directed-polymer free energy

Let $\beta_c$ be the critical inverse temperature and define the free energy by

$$\mathfrak f(\beta):=\lim_{n\to\infty}\frac{1}{n}\log W_n^\beta.$$

Infinite-order criticality conjecture. The free energy satisfies

$$\lim_{u\to0+}\frac{\log\lvert\mathfrak f(\beta_c+u)\rvert}{\log u}=\infty.$$

This predicts that the free-energy curve is flatter than every power law at the transition, analogous to the proposed critical behavior of the relevant Lyapunov exponent. The source notes that the corresponding equivalence and negativity result is known only under an upper-bounded-environment assumption, while this prediction itself is unresolved.