The integer-distances to integer-coordinates embedding conjecture

Let $M\subset \mathbb{Q}^n$ be a set of points such that the distances between any two points of $M$ are integer.

Embedding conjecture. One can find a Euclidean motion $T$ such that

$$TM\subset\mathbb{Z}^n.$$

Equivalently, a finite or infinite rational point set with pairwise integer distances should admit an isometric placement in the integer lattice. The paper states that the conjecture encompasses the presented embedding properties and is equivalent to the divisibility formulation below; its resolution is not given.