Conjecture on the classification of final motions by super-hyperbolic building blocks

Let $N\geq 4$ and consider the final motions of the $N$-body problem as $t\to\infty$. The building blocks are hyperbolic ($\mathcal H$), parabolic ($\mathcal P$), bounded ($\mathcal B$), oscillatory ($\mathcal{OS}$), and super-hyperbolic ($\mathcal{SH}$) motions, where super-hyperbolic means that the system grows super-linearly with respect to time.

Final-motion classification conjecture. For the $N$-body problem with $N\geq 4$, every logically possible combination of

$$\mathcal H,\mathcal P,\mathcal B,\mathcal{OS},\mathcal{SH}$$

is realized by an orbit, and there are no other possibilities.

This conjecture proposes that super-hyperbolic motion is the only new building block needed beyond those in the three-body Chazy classification. The paper states that the conjecture is strongly supported by a theorem of Marshal and Saari, but the classification itself remains open.