Let $x_1 < \cdots < x_n$ and $y_1 < \cdots < y_n$.
Prove that $\det(e^{x_iy_j}) > 0$.
Expanding out the determinant shows that it suffices to prove:
$$\sum_{even} \sum x_iy_{\sigma{(i)}} > \sum_{odd} \sum x_i y_{\sigma{(i)}}$$ Which is kinda like a generalized rearrangement inequality, but I'm not quite sure how to finish it.
Reading the solutions to "positivity of the determinant" answers my original question, but I would like to see if it is possible to have a proof of the above inequality.