Saari's homographic conjecture for homogeneous potentials

Let $N$ point particles have masses $m_k>0$ and positions $\bi{q}_k$, with homogeneous potential

$$U=\frac{1}{\alpha}\sum_{1\le i<j\le N}\frac{m_i m_j}{|\bi{q}_i-\bi{q}_j|^\alpha}$$

for $\alpha\ne0$. Define the moment of inertia and configurational measure by

$$I=\left(\sum_{1\le i<j\le N}m_i m_j|\bi{q}_i-\bi{q}_j|^2\right)\left(\sum_{k=1}^N m_k\right)^{-1},\qquad \mu=\alpha I^{\alpha/2}U.$$

Assume that $\mu$ is not identically constant. Saari's homographic conjecture. If a motion has constant value of $\mu$, then the motion is homographic. This is the extension of Saari's conjecture to arbitrary homogeneous potentials; the paper proves the claim for general-mass planar three-body motion when $\alpha=1$ or $\alpha=2$, while the general statement remains unresolved.