Universal weak-disorder asymptotics for directed-polymer free energy

Let $\beta>0$ denote the disorder strength, let $F(\beta)$ be the directed-polymer free energy, and let $\boldsymbol F(1)$ be the corresponding continuum free energy at unit coupling. Universal directed-polymer free-energy conjecture. For any directed polymer model satisfying Assumption~,

$$\lim_{\beta\downarrow 0} \frac{F(\beta)}{\beta^{\frac{2\alpha}{\alpha-1}}}=\lim_{\delta\downarrow 0}\frac{F(\delta^{\frac{\alpha-1}{2\alpha}})}{\delta}=\boldsymbol F(1).$$

This predicts a universal weak-disorder scaling law governed by the continuum directed-polymer model. The supplied text states the prediction but gives no resolution or supporting theorem for this asymptotic.