Universality conjecture for the random graph of a good polygon configuration

Let $(\mathcal{P}_{n})_{n \geqslant 1}$ be a good sequence of configurations. For each $n$, let $\mathbf{G}_{\mathcal{P}_{n}}$ be the associated random graph, let $\mathbb{G}_{n}$ be the random graph structure of a uniform random labeled map on $n$ edges, and let $\mathbb{G}_{n}^{\mathrm{odd}}$ and $\mathbb{G}_{n}^{\mathrm{even}}$ denote the corresponding laws conditioned on having an odd or even number of vertices, respectively. Write $\epsilon_{n}\in\{\mathrm{even},\mathrm{odd}\}$ for the parity of $n+\#\mathcal{P}_{n}$. Universality conjecture. The total variation distance satisfies

$$\mathrm{d_{TV}}\left(\mathbf{G}_{\mathcal{P}_{n}},\mathbb{G}_{n}^{\epsilon_{n}}\right)\to 0\quad\text{as }n\to\infty.$$

This conjecture asserts that, subject to the parity forced by Euler's formula, the random graph associated with any good sequence of polygon configurations has the same asymptotic law as the graph of a uniform random labeled map. The source presents it as a motivating conjecture; no resolution is supplied here.