Orthogonal symmetry conjecture for Maass cusp forms

Let $\mathcal{B}_N$ be the family of Maass cusp forms of level $N$, let $h_T$ be the weight function defined in the source, let $\phi$ be an even Schwartz function with compactly supported Fourier transform, and suppose that $R\asymp T^2N$. Let $D_1(\mathcal{B}_N,\phi,R;w)$ denote the averaged weighted one-level density, where $w(t_u)$ is the specified weight. Orthogonal symmetry conjecture.

$$\lim_{R\to\infty}D_1(\mathcal{B}_N,\phi,R;w)=\int_{\mathbb{R}}\phi(t)W_{1,\operatorname{SO}}(t)\,dt,$$

where

$$W_{1,\operatorname{SO}}:=1+\frac{1}{2}\delta_0.$$

Equivalently, the symmetry group associated with the family of Maass cusp forms of level $N$ is orthogonal. The conjecture identifies the low-lying-zero statistics of this family with the orthogonal compact-group model; the source presents it as the main question for the family and does not provide a resolution.