Graph-cover characterization conjecture for Bethe partition functions of DE-NFGs

Let $\mathsf{N}$ be a DE-NFG, for which $Z_{\mathrm{B},M}(\mathsf{N})$ denotes the degree-$M$ Bethe partition function and $Z_{\mathrm{B,SPA}}^{*}(\mathsf{N})$ denotes the Bethe partition function associated with the sum-product-algorithm formulation for DE-NFGs.

Graph-cover characterization conjecture. It holds that

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

This conjecture asks whether the graph-cover characterization of the Bethe partition function for S-NFGs extends to DE-NFGs. The right-hand side uses $Z_{\mathrm{B,SPA}}^{*}(\mathsf{N})$, since the ordinary Bethe partition function $Z_{\mathrm{B}}(\mathsf{N})$ is not defined for DE-NFGs; the paper proves the claim for a class of DE-NFGs satisfying a checkable condition.