Coefficientwise Bethe permanent lifting conjecture

Let $\boldsymbol{\theta}$ be a matrix of indeterminates, let $M\in\mathbb{Z}_{>0}$, and let $\tilde{\Psi}_M$ be the set of degree-$M$ graph-covering permutations. For $\tilde{{\bm{P}}}\in\tilde{\Psi}_M$, write $\boldsymbol{\theta}^{\uparrow\tilde{{\bm{P}}}}$ for the associated lifted matrix. Coefficientwise Bethe permanent lifting conjecture. The coefficient of every monomial in

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

is at most the coefficient of the corresponding monomial in $\big(\operatorname{perm}(\boldsymbol{\theta})\big)^M$. Possibly the stronger statement holds with the averaged permanent replaced by $\operatorname{perm}(\boldsymbol{\theta}^{\uparrow\tilde{{\bm{P}}}})$ for every $\tilde{{\bm{P}}}\in\tilde{\Psi}_M$. This coefficientwise version would imply the numerical lifting bound after specializing the indeterminates to non-negative values, but its general status is open.