Inspired by the paper Heronian Tetrahedra Are Lattice Tetrahedra by Susan H. Marshall and Alexander R. Perlis, I started thinking about higher dimensional Heronian simplices.
Heronian simplices are simplices such that the measure of every $n$-dimensional face is an integer. In the case of tetrahedra, this means the side lengths are integers, the areas of the faces are integers, and the volume of the tetrahedra itself is an integer.
However, I read in the paper Lattice Embedding of Heronian Simplices by W. Fred Lunnon that
Sascha Kurz recently completed the enumeration of primitive Heronian simplices to diameter 600,000. He reports 41563542 triangles, 2526 tetrahedra, and 0 pentatope[s].
It seems plausible to me that no Heronian 4-simplices exist—there are just so many constraints! Is there any way to prove this or is there a better heuristic for why this may or may not be the case?