Conjecture on vortex corrections to the effective inverse temperature and free energies

Let $\hat\beta_{\mathrm{eff}}$ be the effective inverse temperature defined in the source, and let $f^{\operatorname{Coul}}(\beta)$, $f^{\operatorname{IV}}(\beta^{-1})$, $f^{\operatorname{Vil}}(\beta)$, and $f^{\operatorname{GFF}}$ denote the corresponding free energies. Vortex-correction conjecture. As $\beta\to\infty$, vortices contribute to the fluctuations so that

$$\lim_{\beta\to\infty}\frac{\log(\beta-\hat\beta_{\mathrm{eff}})}{\beta}=-\pi^2.$$

Moreover, the Coulomb free-energy lower bound is asymptotically sharp:

$$\lim_{\beta\to\infty}\frac{\log f^{\operatorname{Coul}}(\beta)}{\beta}=-\pi^2$$

and

$$-\pi^2=\lim_{\beta\to\infty}\frac{\log\left(f^{\operatorname{IV}}(\beta^{-1})-f^{\operatorname{GFF}}(\beta^{-1})\right)}{\beta}=\lim_{\beta\to\infty}\frac{\log\left(f^{\operatorname{Vil}}(\beta)-f^{\operatorname{GFF}}(\beta)\right)}{\beta}.$$

These predictions quantify the exponentially small effect of vortices at low temperature and are supported in the source by rigorous lower bounds and renormalization-group analysis. The supplied text gives no resolution.