GOE level-spacing conjecture for Liouville quantum gravity

Let $\gamma\in(0,2)$, let $\mu_\gamma$ be the Liouville measure on $D$, and let $(\boldsymbol{\lambda}_j)_{j\ge1}$ be the LQG eigenvalues. Write $F_{\mathrm{GOE}}(x)$ for the GOE level-spacing distribution. GOE spacing conjecture. For each $x\ge0$,

$$\frac1N\sum_{j=1}^N 1_{\{c_\gamma\mu_\gamma(D)(\boldsymbol{\lambda}_{j+1}-\boldsymbol{\lambda}_j)\le x\}} \xrightarrow[N\to\infty]{p}F_{\mathrm{GOE}}(x).$$

The rescaling uses the Weyl-law mean spacing. The conjecture is motivated by the Bohigas–Giannoni–Schmit prediction and is unresolved in the source.