Le Donne's conjecture that Carnot groups have no periodic geodesics

A Carnot group is a connected, simply connected nilpotent Lie group equipped with a bracket-generating left-invariant sub-Riemannian distribution and a left-invariant inner product on that distribution. A periodic geodesic is a geodesic whose trajectory is periodic.

Le Donne's conjecture. Carnot groups do not have periodic geodesics.

The paper notes that its theorem for the jet space $J^k(\mathbb{R},\mathbb{R}^n)$ is a particular case of this conjecture. Its general status is not established in the supplied text.