Let $S$ be the class of all $2$ by $2$ matrices of the form
$$\begin{bmatrix}
    1       & a \\
    a       & 1  
\end{bmatrix},\, |a|\leq 1.$$
Is there a single matrix $M\in S$ such that for any $N\in S$ and all $x>0$ we have
$$\mathbb{P}(||X||_2 \geq x)\geq \mathbb{P}(||Y||_2 \geq x),$$
where $X$ and $Y$ have the Gaussian distribution with covariance matrices $M$ and $N$, respectively?
In other words, is it possible to determine which covariance structure gives the fattest tail behavior?