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Iosif Pinelis
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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:

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Iosif Pinelis
  • 127.8k
  • 8
  • 107
  • 229