Generalized Marchal's conjecture for odd-body polygon and figure-eight choreographies

Let $n$ be an odd number of bodies. Consider the $n$-gon choreography and the $n$-body figure-eight choreography. Two periodic orbits are in the same continuation class if one can be reached from the other by continuous variation of the energy.

Generalized Marchal's conjecture. For any odd number of bodies, the $n$-gon choreography and the $n$-body figure eight are in the same continuation class.

This generalizes the three-body claim and is motivated by numerical continuations for three and five bodies. The paper presents numerical evidence for odd numbers of bodies from $3$ through $15$, but the assertion remains unproved in general.