Applicability of the DMS25Contact construction to realize all Reeb orbits as magnetic geodesics

Determine whether the metric construction developed by Deschamps, Maier, and Stalljohann in "On the contact type conjecture for exact magnetic systems" (arXiv:2508.01113) can, when the Reeb flow of a closed contact manifold (M, α) admits infinitely many periodic orbits, be used to produce a Riemannian metric g for which every Reeb orbit is a magnetic geodesic of the exact magnetic system (M, g, α) with magnetic potential equal to the contact form α.

Background

The paper constructs an infinite-dimensional family of Riemannian metrics G on a closed contact manifold (M, α) such that every constant reparametrization of a Reeb orbit is both a geodesic of (M, g) and a magnetic geodesic of the exact magnetic system (M, g, dα), yielding strong growth results for periodic magnetic geodesics.

In contrast, the prior work [DMS25Contact] provides a different metric construction that is only locally prescribed near Reeb orbits and does not necessarily impose global conditions ensuring that all Reeb orbits are magnetic geodesics. The authors explicitly state uncertainty about whether that construction can be strengthened, in the case of a Reeb flow with infinitely many periodic orbits, to produce a metric making every Reeb orbit a magnetic geodesic of (M, g, α).