Abbondandolo's contact type conjecture for exact magnetic systems

Let $M$ be the configuration manifold, let $L$ be the corresponding magnetic Lagrangian, and let $\Sigma_{\kappa}$ be the energy hypersurface at energy $\kappa$. The lowest Mañé critical value is denoted by $c_u(L)$. Abbondandolo's contact type conjecture. $\Sigma_{\kappa}$ is not of contact type for

$$0 < \kappa < c_u(L).$$

This conjecture concerns the low-energy range below the lowest Mañé critical value, complementing the known fact that energy levels above the strict Mañé critical value are of restricted contact type. The claim remains an open question for exact magnetic systems.