The integer-coordinate conjecture for Heron simplices

A Heron simplex is a simplex whose edge lengths, face areas, and volume are all integers. The conjecture asserts that it can be represented in Euclidean space with integer coordinates.

Integer-coordinate conjecture. Every Heron simplex can be represented with integer coordinates.

The preceding proposition establishes rational coordinates, so the conjecture asks whether the denominators can always be removed by an appropriate representation. It is presented as a special case of the broader embedding conjecture below.