Bethe permanent lifting bound conjecture

Let $\boldsymbol{\theta}$ be a non-negative $n\times n$ matrix. For $M\in\mathbb{Z}_{>0}$, let $\tilde{\Psi}_M$ be the set of degree-$M$ graph-covering permutations and let $\boldsymbol{\theta}^{\uparrow\tilde{{\bm{P}}}}$ denote the matrix lifted according to $\tilde{{\bm{P}}}$. Then $\operatorname{perm}(\boldsymbol{\theta}^{\uparrow\tilde{{\bm{P}}}})$ is defined for every $\tilde{{\bm{P}}}\in\tilde{\Psi}_M$. Bethe permanent lifting bound conjecture. For any $M\in\mathbb{Z}_{>0}$,

$$\Big\langle \! \operatorname{perm}\left(\boldsymbol{\theta}^{\uparrow\tilde{{\bm{P}}}}\right)\!\Big\rangle_{\tilde{{\bm{P}}}\in\tilde{\Psi}_M}\leqslant\big(\operatorname{perm}(\boldsymbol{\theta})\big)^M.$$

Possibly the stronger pointwise inequality

$$\operatorname{perm}\left(\boldsymbol{\theta}^{\uparrow\tilde{{\bm{P}}}}\right)\leqslant\big(\operatorname{perm}(\boldsymbol{\theta})\big)^M$$

holds for every $M\in\mathbb{Z}_{>0}$ and every $\tilde{{\bm{P}}}\in\tilde{\Psi}_M$. The weaker form would imply the claimed upper bound on the Bethe permanent; the conjecture is verified for the all-one matrix and for the matrices studied through finite graph covers, but remains open in general.