Define a *rational $d$-simplex* as a simplex in $\mathbb{R}^d$
such that the measure of all its $k$-dimensional faces, $k \ge 1$, is rational.
So a rational triangle has rational edge lengths and rational area.
A rational tetrahedron has six rational edge lengths, four rational face areas,
and rational volume. And so on. (By scaling these could all be natural numbers.)

. For which $d$ is it known that there are an infinite number of non-similar rational simplices in $\mathbb{R}^d$ ?Q

(The non-similar restriction is to exclude scalings.)
This is in some sense a request to update Richard Guy's *Unsolved Problems In Number Theory*, Problem D22. But I am specifically interested in the boundary between an infinite number of examples, and a finite class of distinct examples.
Is that boundary: *Yes* ($\infty$) for $d \le 3$, and *No* (or unknown?) for $d > 3$? Or is the boundary: $d \le 2$ vs. $d > 2$? Or some other divison?

^{(Image from MathWorld. Face areas $1170, 1800, 1890, 2016$; volume $18144$.) }

** Update**. Answered in the comments by

*j.c.*and

*Moritz Firsching*: Infinite families are known up to $d=3$, but not for $d \ge 4$ (and it seems unlikely even in $d=4$).