Navin-type multiplicative inequality for degree-$M$ Bethe permanents

Let $\bm{\theta}$ be the matrix used to define the degree-$M$ Bethe permanent $\operatorname{perm}_{\mathrm{B},M}(\bm{\theta})$. For any integers $M_{1}\in\mathbb{Z}_{\geq 1}$ and $M_{2}\in\mathbb{Z}_{\geq 1}$,

Navin-type conjecture. It holds that

$$\frac{\bigl(\operatorname{perm}_{\mathrm{B},M_{1}}(\bm{\theta})\bigr)^{M_{1}}\bigl(\operatorname{perm}_{\mathrm{B},M_{2}}(\bm{\theta})\bigr)^{M_{2}}}{\bigl(\operatorname{perm}_{\mathrm{B},M_{1}+M_{2}}(\bm{\theta})\bigr)^{M_{1}+M_{2}}}\geq 1.$$

The paper notes that the case $M_{1}=1$ is proved in an appendix, while the general inequality is presented as a conjecture. It concerns the multiplicative behavior of degree-$M$ Bethe permanents under combining cover degrees.