The GOE level-crossing law for real orthogonal matrices

Let $A$ and $B$ be independently chosen from the real Gaussian orthogonal ensemble $GOE_n^{\mathbb R}$, consisting of real symmetric $n\times n$ matrices with the standard GOE distribution. Their branch-point measure is the empirical measure of the $n(n-1)$ level crossings of the pencil $A+\lambda B$ on $\mathbb{CP}^1$. GOE level-crossing conjecture. For every $n\geq 3$, the branch-point distribution is uniform on $\mathbb{CP}^1$, with affine density

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

The case $n=2$ is stated as a theorem in the source; the conjecture extends that conclusion to all higher dimensions.