Long's elliptic closed-geodesic conjecture for Finsler two-spheres

Let $F$ be a Finsler metric on the two-sphere $S^2$, and let a closed geodesic be a periodic geodesic of $F$. Long's elliptic-geodesic conjecture. There exists at least one elliptic closed geodesic for every Finsler metric on $S^2$. This is presented as a conjecture related to the preceding multiplicity results; the supplied text gives no resolution beyond those results, so its general status remains open.