Canonical bijection between degenerate tetrahedra and planar quadruples

A quadruple of points in the special affine plane determines triangle and Varignon-parallelogram areas. Consider the corresponding quadruple of Euclidean-plane points for which the radius of gyration $R_\mathsf G$ attains its unique minimum subject to preserving those areas. Let degenerate tetrahedra be the tetrahedra defined by the zeros of $\Omega$. Canonical-bijection conjecture. The bijection between affine-plane quadruples and the corresponding minimizing Euclidean-plane quadruples extends to a bijection between the set of degenerate tetrahedra and the set of all quadruples in the Euclidean plane; moreover, this mapping is, or can be chosen to be, canonical, so that the two sets can be identified.