Soules's maximum-eigenvalue conjecture for Schur power matrices

For $A=[a_{i,j}]\in{\cal H}_n$, define the Schur power matrix $\pi(A)$ as the $n!\times n!$ matrix indexed by permutations $\sigma,\tau\in\mathcal{S}_n$, with entries

$$\pi(A)_{\sigma,\tau}=\prod_{t=1}^n a_{\sigma(t),\tau(t)}.$$

Soules's Schur power matrix conjecture. The permanent $\mathop{\rm per} A$ is the maximum eigenvalue of $\pi(A)$. The conjecture is refuted: the source gives a positive-semidefinite counterexample of order five, while noting that the order-four case remains open.