Let $n$ be a positive integer and let $X_n$ be an $n\times n$ random matrix whose entries are iid standard gaussian random variables. I am interested in the distribution of the average singular value of $\frac1{\sqrt{n}}X_n$.

Let $\alpha_n$ be the expected value of the average of the singular values of $\frac1{\sqrt{n}}X_n$. $\alpha_1$ is easy to compute and is equal to $\sqrt{\frac2\pi}$. It is also easy to understand the $n\to\infty$ limit of this distribution because $W_n = \frac1n X_nX_n^T$ is a Wishart Matrix whose eigenvalues will follow, in the limit, the Marchenko-Pastur distribution. Howerever, I am interested in finite $n$ results.

Computer Simulations suggest that $\alpha_n$ increases with $n$. I suspect this might be a known fact that I can't seem to find in the literature, I would really appreciate if someone could point me to the right reference!