The nonexistence conjecture for gentile tetrahedra without right dihedral angles

A tetrahedron without a right dihedral angle is a tetrahedron for which no dihedral angle is exactly $pi/2$. A gentile tetrahedron is a tetrahedron that can be subdivided into finitely many smaller figures, each similar to the original tetrahedron, without requiring the pieces to be mutually congruent.

Non-right-dihedral gentile tetrahedra conjecture. There are no gentile tetrahedra that do not have a dihedral angle of exactly $pi/2$.

This is presented as an even bolder conjecture, extending the acute case to all tetrahedra without a right dihedral angle. It remains open.