The minimum hyperplane-covering conjecture for finite grids

Let $R$ be a ring, and let $A_1,\dots,A_n\subset R$ be nonempty (but possibly infinite) subsets. Let $\mathcal{H}=\{H_j\}_{j\in J}$ be a covering of the grid

$$A=\prod_{i=1}^n A_i$$

by hyperplanes. The minimum hyperplane-covering conjecture. Then

$$\#J\geq\min_{i=1}^n\#A_i.$$

This extends the finite-grid lower bound for hyperplane coverings to possibly infinite coordinate sets; the supplied text does not indicate whether the assertion is known in this generality.