My calculations confirm your result for $n=3$. However, for $n=4$, the derivative of $\pi(1)/\pi(2)$ in $x_3$ at the point $(x_1,x_2,x_3,x_4)=(1,2,3,4)$ is strictly positive, which disproves the conjecture for $n=4$.
Below are my calculations with Mathematica:
