The real sample covariance analogue of the largest-eigenvalue phase transition

Let the sample covariance matrix be real rather than complex, with distinguished covariance eigenvalues $\ell_j$ and aspect-ratio limit parameter $\gamma$ as in Theorem 1. The theorem describes the phase transition and scaling of the largest eigenvalue in the complex sample covariance model.

Real sample covariance conjecture. For real sample covariance, Theorem 1 should still hold, with different limiting distributions but the same scaling. In particular, the critical value of a distinguished eigenvalue $\ell_j$ of the covariance matrix is expected to be

$$1+\gamma^{-1}.$$

The conjecture proposes that changing from complex to real samples changes the limiting distributions but not the scaling or the location of the phase transition. The source gives no resolution, so the conjecture remains open.