Bethe permanent kernel conjecture

Let $n$ be a positive integer, let $\mathcal{X}$ be a set endowed with a kernel $\kappa$, and let $X=(x_1,\ldots,x_n)\in\mathcal{X}^n$ and $Y=(y_1,\ldots,y_n)\in\mathcal{X}^n$. Define the Bethe-permanent kernel by

$$\kappa_{\operatorname{perm}_{\mathrm{B}}}(X,Y)=\operatorname{perm}_{\mathrm{B}}\Big(\big[\kappa(x_i,y_j)\big]_{1\leqslant i,j\leqslant n}\Big).$$

Huang–Jebara Bethe permanent kernel conjecture. The function $\kappa_{\operatorname{perm}_{\mathrm{B}}}$ is a positive definite kernel on $\mathcal{X}^n\times\mathcal{X}^n$. The claim is presented as a conjecture based on Cuturi's result; no resolution is given in the supplied text.