Bapat–Sunder positive-definite-function conjecture

Let $\mathcal{S}_n$ be the symmetric group and let $c:\mathcal{S}_n\to\mathbb{C}$ satisfy

$$\sum_{\sigma,\tau\in\mathcal{S}_n}\overline{x(\tau)}\,c(\sigma\tau^{-1})\,x(\sigma)\geqslant0$$

for every complex-valued function $x$ on $\mathcal{S}_n$. For $A=[a_{i,j}]\in{\cal H}_n$, Bapat–Sunder's reformulated conjecture.

$$c({\bf1}_n)\mathop{\rm per} A\geqslant\sum_{\sigma\in\mathcal{S}_n}c(\sigma)\prod_{i=1}^n a_{i,\sigma(i)}.$$

It is equivalent to the Schur power matrix conjecture and is therefore refuted by the counterexamples described in the source.