Uniqueness of the equal-eigenvalue balanced configuration in the 3-body problem in four dimensions

Let a balanced configuration of the three-body problem in $\mathbb{R}^4$ have angular momentum $L$ and eigenvalues $\mu_1$ and $\mu_2$. Uniqueness conjecture. For any choice of masses, a unique balanced configuration allows a relative-equilibrium motion with given angular momentum $L$ and equal eigenvalues

$$\mu_1=\mu_2.$$

The claim would imply that short balanced families do not reach $k=1/4$ and, together with the proposition on cusps, that each short family has at least one cusp; the source presents existence and uniqueness as apparent rather than established.