Saari's original conjecture for the Newtonian $N$-body problem

Consider the Newtonian $N$-body problem, with potential corresponding to $\alpha=1$, and let $I$ be the moment of inertia. A motion has constant moment of inertia when $I$ is constant in time. Saari's original conjecture. If a motion has constant moment of inertia, then it is a relative equilibrium: the bodies rotate about their centre of mass as though fixed in a rigid body. The conjecture was proved for the three-body problem with general masses in spatial dimension at least two, but the statement as presented for the general $N$-body problem is not resolved here.