Potential-theoretic form of the Hermitian level-crossing limit

Let $\lambda\in\mathbb C$ be an affine coordinate on $\mathbb{CP}^1$, let

$$Y=\frac{2\operatorname{Im}\lambda}{1+|\lambda|^2},$$

and let $G(q)$ denote the logarithmic energy of the elliptic law with elliptic parameter $\sqrt q$. Let $\mathbb{RP}^1\subset\mathbb{CP}^1$ be the real projective line. Hermitian potential-theoretic limit conjecture. The universal Hermitian limiting measure is the $SO(2)$-invariant measure which, away from $\mathbb{RP}^1$, has density

$$\frac{1}{2\pi}\Delta_\lambda\left[\frac12\log(1+|\lambda|^2)+G(1-Y^2)\right]dx\,dy,$$

and which has no atom or singular component on $\mathbb{RP}^1$. This gives a potential-theoretic description of the same universal measure expected for GUE and general Hermitian Wigner pencils.