The large-grid hypothesis for forthcoming translation results

Let $\mathcal{G}_g$ be a grid graph with $D$ dimensions, whose dimension sizes are given by $\mathbf{d}[i]$ for $i\in\{1,\ldots,D\}$. The preceding large-grid hypothesis assumes

$$\mathbf{d}[D]\geq 3\quad\text{and}\quad \forall i\in\{1,\ldots,D-1\}:\ \mathbf{d}[i]\geq 2+2\prod_{j=i+1}^{D}\mathbf{d}[j].$$

Large-grid conjecture. The forthcoming results apply for grid graphs such that

$$\forall i\in\{1,\ldots,D\}:\ \mathbf{d}[i]\geq 6.$$

This is presented as a conjectural condition preceding results about Euclidean translations by standard basis vectors or their negatives being pseudo-minimal on grid graphs. The source does not provide evidence that this claim has been resolved.