Berry–Tabor conjecture for Laplacian spacings on Diophantine irrational flat tori

Let ${\mathcal L}_0$ be a Diophantine irrational lattice. Write $n_j$ for the distinct Laplacian eigenvalues, let $\delta_j=n_j-n_{j-1}>0$, let

$$N_{{\mathcal L}_0}(x)=\#\{n_j\leq x\},$$

and define the mean-normalized spacings by $\hat{\delta}_j=\delta_j/\langle\delta_j\rangle_x$ for $n_j\leq x$. Berry–Tabor conjecture. The spacings have a Poisson distribution of mean $1$ as $x\to\infty$: for every $h\in C^\infty(\mathbb R_+)$,

$$\lim_{x\to\infty}\frac{1}{N_{{\mathcal L}_0}(x)}\sum_{n_j\leq x}h(\hat{\delta}_j)=\int_0^\infty h(s)e^{-s}\,ds.$$

This is the flat two-dimensional torus case of the Berry–Tabor conjecture for generic classically integrable quantum systems. The source presents it as an expected asymptotic and gives no evidence of a resolution, so it remains open here.