Conjecture on contact type for positive magnetic curvature

Let $(S^2,g,\sigma)$ be a symplectic magnetic system. For $m>0$, define the magnetic curvature on the energy level $\Sigma_m$ by

$$K_m:=m^2K+1.$$

Assume that $K_m$ is positive. Contact-type conjecture. Then $\Sigma_m$ is of contact type. This conjecture concerns whether positivity of the magnetic curvature suffices for the contact property on every energy level; numerical computations for ellipsoids support it, while the general statement remains open.