The prime divisibility embedding conjecture

Let $p$ be a prime. Let $M\subset\mathbb{Z}^n$ be a set of points such that the distances between any two points of $M$ are integer and divisible by $p$.

Prime divisibility embedding conjecture. There is a Euclidean motion $T$ such that

$$TM\subset(p\mathbb{Z})^n.$$

This is the prime-divisor version of the divisibility embedding conjecture. The source gives no resolution of this claim.