As in the object, I'm looking at the case where $x \in \mathbb R^d$ is a generic vector, and $AA^\top \in \mathbb R^{d \times d}$ is a p.s.d. matrix.

I'm investigating the following inequality

$x^\top e^{-xx^\top - AA^\top} x \leq x^\top e^{-xx^\top} x,$

where I'm considering matrix exponentials. In general, it is not true that if $BB^\top \geq CC^\top$ in p.s.d. sense, then we have $e^{BB^\top} \geq e^{CC^\top}$. However, here, I'm looking for something weaker, but I'm struggling to prove it / found a counter example!