I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance matrix.

Specifically, let $X \sim N(\mathbf{\mu}, \Sigma)$ where $\mu \in \mathbb{R}^d$ and $\Sigma$ is an arbitrary PSD matrix. I am hoping to bound the distance between $\|x\|_\infty$ and $\mathbb{E}\,\|X\|_\infty$, in other words$$ \mathbb{E}_{x \sim N(\mu,\Sigma)}\,\left|\left(\max_{i=1, \ldots, d} x_i\right) - \left(\mathbb{E}_{x \sim N(\mu,\Sigma)}\,[\max_{i=1,\ldots,d} X_i]\right) \right|^2. $$

All references I've come across seem to handle the special case of $X$ being centered (zero mean) and usually with spherical covariance matrix:

- https://www.math.ucla.edu/~biskup/PIMS/PDFs/lecture6.pdf
- https://people.eecs.berkeley.edu/~stephentu/blog/probability-theory/2017/10/16/upper-and-lower-tails-gaussian-maxima.html
- Concentration inequality for maximum of gaussians

Any references or suggestions are gladly welcomed.