Gromov--Hausdorff coupling conjecture for unicellular maps and hyperbolic surfaces

Let $n_g$ satisfy $g=o(n_g)$ as $g\to\infty$. Let $\mathbf U_{n_g,g}$ be a uniformly random unicellular map of genus $g$ and size $n_g$, equipped with its graph metric rescaled by $L_{n_g}^{-1}$, and let $\mathbf S_g$ be a random hyperbolic surface of genus $g$ sampled according to the Weil--Petersson probability measure. Write $d_{\mathrm{GH}}$ for the Gromov--Hausdorff distance between metric spaces.

Gromov--Hausdorff coupling conjecture. The variables $\mathbf U_{n_g,g}$ and $\mathbf S_g$ can be coupled so that

$$d_{\mathrm{GH}}\left(\mathbf U_{n_g,g},\mathbf S_g\right)\xrightarrow[g\to\infty]{}0$$

in probability.

This is the precise form of the paper's proposed correspondence between unicellular maps and Weil--Petersson random hyperbolic surfaces. It is motivated by matching limiting statistics of short simple cycles and simple closed geodesics; no proof or resolution is supplied in the source.