The divisibility embedding conjecture

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 a number $k$. In this setting, $M$ is congruent to a set after scaling by $k$ if there is a set $N\subset\mathbb{Z}^n$ such that $kN$ is congruent to $M$.

Divisibility embedding conjecture. There is a set $N\subset\mathbb{Z}^n$ such that $k\cdot N$ is congruent to $M$.

The paper states that this formulation is equivalent to the integer-distances to integer-coordinates embedding conjecture. No resolution is supplied.