Centroid characterization when central and reference tetrahedra are similar

Let a reference tetrahedron and its associated central tetrahedron be given, and suppose the two tetrahedra are similar, meaning that corresponding edge lengths are proportional. Let the centroid be the common point determined by equal barycentric coordinates. Centroid conjecture. If the central tetrahedron is similar to the reference tetrahedron, then the center must be the centroid. The paper presents this claim as data-supported and formally unproved.