Hermitian Wigner universality for level crossings

Fix a real-valued probability distribution $\mu$ with mean zero and finite moments, and a complex-valued probability distribution $\nu$ with mean zero, unit variance, finite moments, and Hermitian symmetry $h_{ji}=\overline{h_{ij}}$. Let $\mathcal H E_{\mu,\nu,n}$ be the Wigner ensemble of Hermitian $n\times n$ matrices whose diagonal entries are i.i.d. with law $\mu$ and whose upper-triangular off-diagonal entries are i.i.d. with law $\nu$. Hermitian Wigner universality conjecture. If $A_n$ and $B_n$ are independent matrices from $\mathcal H E_{\mu,\nu,n}$, normalized in the usual Wigner scaling, then the empirical level-crossing measures of $A_n+\lambda B_n$ converge as $n\to\infty$ to the same probability measure $\Psi$ as in the GUE limiting-density conjecture. Thus the limiting law is universal within the Hermitian symmetry class and independent of the particular laws $\mu$ and $\nu$. The conjecture is the non-Gaussian universality extension of the Hermitian Gaussian result.