Asymptotic distribution of singular values for matrices in a spherical ensemble

We consider the asymptotic behavior of the singular values of a so-called spherical ensemble of random matrices of large dimension. These are matrices of the form $\mathbf X\mathbf Y^{-1}$, where $\mathbf X$ and $\mathbf Y$ are independent matrices of dimension $n\times n$ whose symmetric entries have correlation coefficient $\rho$. We show that the limit distribution of the singular values is independent of the correlation coefficient and has the density
$$ p(x)=\frac1{\pi\sqrt x(1+x)}\mathbb I\{x>0\}, $$
where $\mathbb I\{A\}$ stands for the indicator of an event $A$.