Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as

$$\rho(M)=\frac{n \sum_i \sigma_i^4}{\left(\sum_i \sigma_i^2\right)^2}$$

Take square matrices $A$, $B$ with entries sampled IID from distributions $\mathcal{D}_1$, $\mathcal{D}_2$ respectively. The following appears to hold empirically for standard normal $\mathcal{D}_1$ and symmetric uniform $\mathcal{D}_2$

$$\rho(AB)\approx \rho(A)+\rho(B)-1$$

Is this a well-known result? Any pointers to the literature appreciated!