Generalised Saari's conjecture for homogeneous potentials

Consider the homogeneous potential with exponent $\alpha$ as above, and the moment of inertia $I$. The exceptional exponents are $\alpha=-2$ and $\alpha=2$; the equal-mass rectilinear three-body problem with $\alpha=-4$ is also excluded. Generalised Saari's conjecture. Saari's original conjecture extends to homogeneous potentials with $\alpha\ne-2$ and $\alpha\ne2$, subject to the stated rectilinear equal-mass three-body exclusion. For $\alpha\ne2$, the paper explains that this generalised conjecture is contained in the homographic conjecture; the general claim is not established in the source.