Does a simplex with equal altitudes have to be equilateral?

Question

Consider a simplex in $\mathbb{R}^d$. Assume that all its altitudes have the same length. Does it necessarily mean that the simplex is equilateral, i. e. all distances between its vertices are equal too?

Answer 1

Well, it's obviously wrong. Indeed, we can get a 3-dimensional tetrahedron where edges are divisible into pairs, and each of them contains 2 non-intersecting edges with the same length. Then each facet of the tetrahedron has the same square $S$, and it easily follows from this that each altitude has a length equal to $\frac{3\cdot Volume}{S}$, but the tetrahedron isn't regular.