The rigidity conjecture for partially rigid motions in the n-body problem

Let a solution of the $n$-body problem in $\mathbb{R}^d$ be called partially rigid if at least one, but not all, of the mutual distances remains constant. A relative equilibrium is a solution for which all mutual distances remain constant.

Rigidity conjecture. There are no partially rigid solutions of the $n$-body problem in $\mathbb{\mathbb{R}}^d$. Equivalently, the only solutions such that one of the mutual distances remains constant are the relative equilibrium solutions.

The conjecture is false if even one of the masses is zero. Its status for positive masses is not established by the supplied statement.