Precise transition for complex symmetric Gaussian-product matrices

Characterize the precise location of the transition between Gaussian behavior and non-Gaussian behavior for the complex symmetric random matrices $GG^T/\sqrt{K}$, where $G$ is an $N\times K$ matrix of independent standard complex Gaussian entries.

Background

The paper proves that the total variation distance between the symmetric Gaussian ensemble and the normalized complex product ensemble $GG^T/\sqrt{K}$ is $O(N/\sqrt{K})$, and therefore tends to zero when $N=o(\sqrt{K})$. The authors compare this range with the known $\Theta(K^{1/3})$ transition for GOE behavior of real Wishart matrices. They leave unresolved the corresponding sharp transition for the complex symmetric product ensemble considered in the paper.