Uniqueness of power-law hyperbolic centers in isodynamic tetrahedra

Let a tetrahedron be isodynamic, and let a center function assign a point on each face. A center function is hyperbolic when the cevians to the corresponding face centers form a hyperbolic group. Uniqueness conjecture. The only hyperbolic center functions for an isodynamic tetrahedron are

$$a^r g(b,c),$$

where $g$ is a symmetric homogeneous function and $r$ is arbitrary. The preceding theorem establishes that every function of this form is hyperbolic; the conjecture asserts that no others occur.