General real i.i.d. level-crossing law

Let $A_n=(a_{ij})$ and $B_n=(b_{ij})$ be independent real $n\times n$ matrices with i.i.d. entries satisfying

$$\mathbb E a_{11}=\mathbb E b_{11}=0,\qquad \mathbb E a_{11}^2=\mathbb E b_{11}^2=1,$$

and assume a standard moment condition strong enough to imply the circular law and the corresponding local no-clustering estimates. Let $\mu_n^{LC}$ be the empirical measure of the $n(n-1)$ level crossings of $A_n+\lambda B_n$, counted with multiplicity and normalized to have total mass one. General real i.i.d. level-crossing conjecture. As $n\to\infty$,

$$\mu_n^{LC}\Longrightarrow\frac{dx\,dy}{\pi(1+|\lambda|^2)^2}$$

in probability on $\mathbb{CP}^1$. Equivalently, the limiting law is the uniform spherical measure. The source explains that possible concentration near $\mathbb{RP}^1$ is an additional issue, while a conditional theorem proves the conclusion under a no-concentration hypothesis.