The zero forcing lower bound for Cartesian products

Let $G$ and $G'$ be graphs, each containing an edge. The Cartesian-product zero forcing conjecture.

$$\operatorname{Z}(G\Box G')\geq \operatorname{Z}(G)\operatorname{Z}(G')+1.$$

The conjecture would extend the sharp lower bound known when the maximum nullity equals the zero forcing number for both factors to arbitrary graphs containing an edge. Together with the standard upper bound for Cartesian products, it would give a general Vizing-like constraint on the zero forcing number.