Let $X=(X\_1,\dots,X\_n)$ and $Y=(Y\_1,\dots,Y\_n)$ be centered Gaussian vectors 
with variance matrix $\Gamma\_X$ and $\Gamma\_Y$. We assume that the matrix 
$\Gamma\_Y-\Gamma\_X$ is positive definite. Is it possible to prove that 
$$ \forall x \ge 0, \mathbb{P}(\max \lbrace |Y\_1|,\dots,|Y\_n| \rbrace \ge x) \ge \mathbb{P}(\max \lbrace |X\_1|,\dots,|X\_n|\rbrace \ge x) .$$
Thank you. 



Inequality for the maximum of Gaussian variables
I think that what you are writing is precisely Anderson's correlation inequality, see "T. W. Anderson. The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc., 6(2):170–176, 1955."


