The independent-set polynomial cover inequality for bipartite graphs

Let $G=(V,E)$ be a finite bipartite graph, let $\tilde{G}$ be its $M$-cover, and let $p(G)=\sum_{I\text{ independent}}\prod_{v\in I}x_v$ be its multivariate independent-set polynomial. Extend the natural projection from $\tilde{G}$ to $G$ to a ring homomorphism $\Pi$ by mapping each indeterminate $x_v$ to $x_{\Pi(v)}$. For polynomials with nonnegative coefficients, write $f\preceq g$ when every monomial coefficient of $f$ is at most the corresponding coefficient of $g$. Independent-set polynomial cover inequality. For any bipartite graph $G$ and its $M$-cover $\tilde{G}$, one has

$$\Pi\bigl(p(\tilde{G})\bigr)\preceq p(G)^M.$$

The inequality compares the projected independent sets of a cover with the independent sets in $M$ disjoint copies of the base graph. It was checked computationally for many examples, but no resolution is supplied here.