Graph-distance scaling limit conjecture for planar maps

Let $a\from (2,5/2)$, let $\bigl(f_i\bigr)_{i\geq 1}$ be the sampled marked vertices or faces, let $d^\dagger_\mathrm{gr}$ denote the graph distance, and let $\bigl(w_i\bigr)_{i\geq 1}$ be the corresponding points of $(\mathcal{D}_a,d_{\mathcal{D}_a})$. Write $c_a$, $p_{\bf q}$, and $e_{\bf q}$ for the model-dependent constants, and interpret the convergence in the product topology. Graph-distance scaling-limit conjecture. Under $\mathbb{P}^{(\ell)}$,

$$\left(\ell^{2-a}d^\dagger_\mathrm{gr}(f_i,f_j)\right)_{i,j\geq 1}\mathop{\longrightarrow}\limits_{\ell\to\infty}^{(\mathrm{d})}\left(\frac{1+e_{\bf q}}{2c_ap_{\bf q}}d_{\mathcal{D}_a}(w_i,w_j)\right)_{i,j\geq 1}.$$

Moreover, $\mathcal{D}_a$ has a compact completion, and this convergence of metric spaces also holds in the Gromov–Hausdorff sense for that completion. The claim concerns the conjectural scaling limit of graph distances and the compactness of the resulting continuum metric space; the supplied text gives no resolution evidence.