Possible non-universal corrections for constrained real ensembles

Consider real matrix ensembles with additional linear constraints, including real symmetric or other structured real matrices. Let $\rho(\lambda)\,dx\,dy$ denote an absolutely continuous limiting level-crossing measure on $\mathbb{CP}^1$ that is invariant under the natural $SO(2)$-action. Constrained-real correction conjecture. Such a density should have the form

$$\rho(\lambda)\,dx\,dy=h\!\left(\frac{|1+\lambda^2|^2}{(1+|\lambda|^2)^2}\right)\frac{dx\,dy}{\pi(1+|\lambda|^2)^2},$$

where $h$ is non-negative and normalized so that the total mass is one; in the full real i.i.d. case, $h\equiv1$. The source presents this as an expected form rather than an established universality statement, reflecting possible non-uniform behavior caused by real symmetry constraints.