Existence of the limiting GUE level-crossing density

For each $n$, let $\mathcal P_n^{GUE}(\theta,\phi)$ denote the level-crossing density for a pencil of independent GUE matrices, expressed in spherical coordinates $(\theta,\phi)$ on $\mathbb{CP}^1$. The source notes that these densities are independent of the azimuthal angle $\phi$. GUE limiting-density conjecture. There exists a limiting density

$$\Psi(\theta,\phi):=\lim_{n\to\infty}\mathcal P_n^{GUE}(\theta,\phi).$$

The conjecture concerns existence of the large-$n$ limit; the source does not provide an exact formula for this limit.