Free-energy asymptotics conjecture for directed polymers on fractal graphs

Let $F_q(\beta)$ and $F_a(\beta)$ denote the quenched and annealed free energies of the directed polymer, respectively, let $d_s$ be the spectral dimension of the underlying fractal graph, and let $\mathcal{Z}_T$ be the limiting random variable from the preceding scaling-limit conjecture. Free-energy asymptotics conjecture. Under the assumption in the scaling-limit conjecture,

$$\lim_{\beta\to 0}\beta^{-\frac{4}{2-d_s}}\bigl(F_q(\beta)-F_a(\beta)\bigr)=\lim_{T\to\infty}\frac{1}{T}\mathcal{P}\left[\log\mathcal{Z}_T\right].$$

The claim identifies the weak-disorder free-energy gap with the long-time growth rate of the limiting stochastic heat equation. It expresses the expected interchange of the small-disorder and large-time limits, but the supplied text gives no resolution.