The divisibility and scaling conjecture for integral point sets

Let $\mathcal{P}\subset\mathbb{Z}^m$ be a finite set of points such that the distances between any two points of $\mathcal{P}$ are integers divisible by an integer $k$. Scaling conjecture. There is a set $\mathcal{P}'\subset\mathbb{Z}^m$ such that $\mathcal{P}'\cdot k$, the set $\mathcal{P}'$ scaled by a factor $k$, is congruent to $\mathcal{P}$. This is presented alongside the embedding conjecture as a tool for placing integral-distance configurations in an integer grid; its general-dimensional status is not resolved in the supplied text.