Growth-rate classification conjecture for sub-superhyperbolic $N$-body motions

Let $N\geq 4$ and let $I(t)=\sum m_i|Q_i(t)|^2$ be the moment of inertia of the system. Consider an orbit satisfying

$$\sqrt{I(t)}=O(t^{2/3})\quad\text{as }t\to\infty.$$

Growth-rate classification conjecture. Every such orbit is a combination of $\mathcal P$, $\mathcal B$, and $\mathcal{OS}$. Conversely, every logically possible combination of $\mathcal P$, $\mathcal B$, and $\mathcal{OS}$ is realized by an orbit satisfying $\sqrt{I(t)}=O(t^{2/3})$ as $t\to\infty$.

This conjecture is presented as a reduction of the broader final-motion classification, using the stated Marshal–Saari dichotomy. The source gives no resolution, so both the classification and its converse remain open.