For graphs $G,H$ it is possible to add $(|V(G)|-γ(G))(|V(H)|-γ(H))$ edges to the Cartesian product $G\mathbin{\square}H$ such that a minimal dominating set $D$ of size $γ(G)γ(H)$ emerges. We hypothesize that $D$ is also a minimum dominating set for the resulting graph and show that this implies Vizing's conjecture.