Graph-cover characterization of the Bethe partition function for double-edge factor graphs

Let $\mathsf{N}$ be a double-edge normal factor graph (DE-NFG). For each integer $M\geq 1$, let $Z_{\mathrm{B},M}(\mathsf{N})$ denote the degree-$M$ Bethe partition function, defined as the $M$th root of the arithmetic mean of the partition functions over all $M$-covers of $\mathsf{N}$. Let $Z_{\mathrm{B,SPA}}^{*}(\mathsf{N})$ denote the sum-product-algorithm fixed-point-based Bethe approximation of the partition function. Graph-cover characterization conjecture. For every DE-NFG $\mathsf{N}$,

$$\limsup_{M\to\infty} Z_{\mathrm{B},M}(\mathsf{N})=Z_{\mathrm{B,SPA}}^{*}(\mathsf{N}).$$

The analogous characterization is known for standard factor graphs, while for DE-NFGs the paper proves the equality under an easily checkable condition and conjectures that it holds more generally.