Pseudocovariance-independence conjecture for GOE level crossings

Let $A_n$ and $B_n$ be independent GOE matrices, and write the normalized complex Gaussian pencil as $\widehat C_n(\lambda)=(A_n+\lambda B_n)/\sqrt{n(1+|\lambda|^2)}$. Its entry pseudocovariance is

$$\tau(\lambda)=\frac{1+\lambda^2}{1+|\lambda|^2},\qquad q(\lambda)=|\tau(\lambda)|^2.$$

Let $\Delta_n(\lambda)$ denote the discriminant of the pencil and let $c_n$ be independent of $\lambda$. Pseudocovariance-independence conjecture. The pseudocovariance parameter does not contribute to the leading asymptotics:

$$\frac{1}{n(n-1)}\log|\Delta_n(\lambda)|=\frac12\log(1+|\lambda|^2)+c_n+o(1)$$

in probability in $L^1_{\rm loc}(\mathbb C)$. This would imply the uniform level-crossing law for GOE pencils; the unresolved issue is whether the pseudocovariance affects the leading logarithmic energy.