  Let $n\geq 2$ and $(X_i,1\leq i\leq n)$ be a centered Gaussian random vector.
The Gaussian minimum conjecture says that $E\left(\min_{1\leq i\leq
n}|X_i|\right)\geq E\left(\min_{1\leq i\leq n}|Y_i|\right)$, where
$Y_1,\ldots,Y_n$ are independent centered Gaussian random variables with
$E(X_i^2)=E(Y_i^2)$ for any $i=1,\ldots,n$. In this note, we will show that
this conjecture holds if and only if $n=2$.
