The list-homomorphism product conjecture

Let $G$ and $H$ be any graphs and let ${\cal L}$ be any list set. For each edge $uv\in E(G)$, using the notation of the weighted bipartite-graph product conjecture, let ${\cal L}^{uv}$ be the list set on $K_{d(u),d(v)}$ given by

$$L^{u,v}(w_j(u,v))=L(n_j(v)),\qquad L^{u,v}(z_j(u,v))=L(n_j(u)).$$

List-homomorphism product conjecture. Then

$$|\operatorname{Hom}^{\cal L}(G,H)|\leq\prod_{uv\in E(G)}|\operatorname{Hom}^{{\cal L}^{uv}}(K_{d(u),d(v)})|^{\frac{1}{d(u)d(v)}}.$$

This conjecture is stated as a list-homomorphism analogue from which the preceding weighted conjecture would follow. Its status is not resolved in the supplied source.