Critical inverse-temperature scaling conjecture for the discrete Gaussian free field Gibbs measure

Let $\mathfrak D$ be the class of domains in the source, let $D_N$ be the corresponding lattice approximation, let $h_z$ be the discrete Gaussian free field, let $m_N$ be its centering, let $\beta_{\mathrm c}$ be the critical inverse temperature, and let $Z^D$ and $\widehat Z^D$ be the limiting random measures defined in the preceding results. Critical scaling conjecture. There is $c\in(0,\infty)$ such that for each $D\in\mathfrak D$,

$$\sqrt{\log N}\sum_{z\in D_N}e^{\beta_{\mathrm c}(h_z-m_N)}\delta_{z/N}(\mathrm d x)\underset{N\to\infty}{\overset{\mathrm{law}}\longrightarrow}c\,Z^D(\mathrm d x).$$

In particular,

$$\sum_{z\in D_N}\mu_{\beta_{\mathrm c},N}^D(\{z\})\delta_{z/N}(\mathrm d x)\underset{N\to\infty}{\overset{\mathrm{law}}\longrightarrow}\widehat Z^D(\mathrm d x).$$

The source expects versions of the limits at the critical inverse temperature $\beta=\beta_{\mathrm c}$, extending the results established away from criticality; the asserted scaling and convergence remain unproved there.