The weighted bipartite-graph product conjecture

Let $G$ be any graph and let $W$ be any collection of weights on $G$. For each $u\in V(G)$, let $\{n_1(u),\ldots,n_{d(u)}(u)\}$ be the set of neighbours of $u$. For each edge $uv\in E(G)$, label the degree-$d(u)$ vertices of $K_{d(u),d(v)}$ by $w_1(u,v),\ldots,w_{d(v)}(u,v)$ and the degree-$d(v)$ vertices by $z_1(u,v),\ldots,z_{d(u)}(u,v)$. Let $W^{uv}$ be the collection of weights on $K_{d(u),d(v)}$ given by

$$\lambda^{u,v}_{i,w_j(u,v)}=\lambda_{i,n_j(v)},\qquad \lambda^{u,v}_{i,z_j(u,v)}=\lambda_{i,n_j(u)},\qquad \lambda^{u,v}_{ij,w_j(u,v)z_k(u,v)}=\lambda_{ij,n_j(v)n_k(u)}.$$

Weighted bipartite-graph product conjecture. Then

$$Z^W(G)\leq\prod_{uv\in E(G)}Z^{W^{uv}}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}}.$$

The conjecture is proposed as the weighted analogue of Kahn's independent-set conjecture and is presented as an extension of a preceding theorem. Its status is not resolved in the supplied source.