Suppose we are given a finite family of points $p\_1,...,p\_n\in \Bbb R^d$, so that any two points have a rational distance square, that is,

$$\|p\_i-p\_j\|^2\in\Bbb Q,\quad\text{for all $i,j\in\{1,...,n\}$}.$$

Is it know whether these points can be isometrically embedded into $\Bbb Q^{D}$ for some sufficiently large $D\ge d$? That is, are there $q\_1,...,q\_n\in\smash{\Bbb Q^D}$ so that

$$\|p\_i-p\_j\|=\|q\_i-q\_j\|,\quad\text{for all $i,j\in\{1,...,n\}$}?$$

If *yes*, what are reasonable lower/upper bounds for $D$?