Unique critical points on balanced families in the 3-body problem in four dimensions

Consider a balanced family in the three-body problem in $\mathbb{R}^4$, with functions $h$ and $k$ defined along the family. Critical-point conjecture. On each balanced family, the functions $h$ and $k$ have exactly one critical point each. The source says this would imply that there is exactly one cusp on each short family and no cusps on the long family, but gives no proof.