Let $X_1, \ldots, X_n$ be jointly Gaussian, each of which is marginally distributed as a standard Gaussian $N(0,1)$. It is well known that $\max |X_i|$ achieves the maximum in the stochastic sense if $X_1, \ldots, X_n$ are independent. To be more precise, $\mathbb{P}(\max|X_i| \le c)$ is minimized in the case of iid $X_1, \ldots, X_n$ for any constant $c > 0$. This fact can be directly seen from the Gaussian correlation conjecture (now a theorem), which was recently proved by Thomas Royen.

**Question** Are there similar results concerning the second largest $|X_i|$, the third largest, etc?

For example, is it true that the second largest $|X_i|$ attains the maximum if $X_1, \ldots, X_n$ are independent?