Large-genus Brownian-surface geodesic length spectrum conjecture

Let $\mathbf{S}_{g}$ be the Brownian surface of genus $g$ with no boundary introduced in Bettinelli (2022). Define $\Xi_g$ to be the multiset of lengths of closed geodesics on $\mathbf{S}_g$, in the sense of shortest closed paths in their homotopy class. Let $c>0$ be an absolute constant, and let $\mathcal{P}$ be a Poisson point process on $\mathbb{R}_{+}$ with intensity

$$\frac{\cosh t-1}{t}\,\mathrm{d}t.$$

Large-genus Brownian-surface geodesic length spectrum conjecture. As $g\to\infty$, the random point process $c\,\Xi_g$ converges in distribution to $\mathcal{P}$:

$$c\,\Xi_g\xrightarrow[g\to\infty]{\mathrm{(d)}}\mathcal{P}.$$

Here $\Xi_g$ is regarded as a random point process on $[0,\infty)$. This conjecture proposes that the limiting Poisson point process found for large-genus random hyperbolic surfaces also describes the rescaled closed-geodesic length spectrum of Brownian surfaces; its resolution is not given in the source.