Isoscelesness of the reference tetrahedron from its central tetrahedron

Let a reference tetrahedron and its associated central tetrahedron be given. A tetrahedron is isosceles when it has the corresponding symmetry/equality of edge lengths intended in the paper. Isosceles central-tetrahedron conjecture. If the central tetrahedron is isosceles, then the reference tetrahedron is isosceles. The paper places this claim among miscellaneous conjectures supported by data but without formal proofs.