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 $$\mathbb{P}(||Z_1||_2 \geq x)\leq \mathbb{P}(||Z_2||_2 \geq x),$$ where $Z_1$ and $Z_2$ have the Gaussian distribution with covariance matrices $N$ and $M$, respectively? In other words, is it possible to determine which covariance structure gives the worst tail behavior?

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?