The rational integral-distance embedding conjecture

Let $\mathcal{P}\subset\mathbb{Q}^m$ be a finite set of points such that the distances between any two points of $\mathcal{P}$ are integers. Embedding conjecture. There is a Euclidean motion $T$ such that

$$T(\mathcal{P})\subset\mathbb{Z}^m.$$

This is stated as a conjecture in general dimension and is a theorem for dimension $m=2$, which is why it can be used to embed Heronian triangles in the integer grid.