The two-or-infinity conjecture for prime closed geodesics on the two-sphere

Let $F$ be a Finsler metric on $S^2$, and call a closed geodesic prime if it is not an iterate of a shorter closed geodesic. Two-or-infinity conjecture for Finsler metrics. Every Finsler metric on $S^2$ has either two or infinitely many prime closed geodesics. This is the Finsler-geometric analogue of the two-or-infinity prediction for Reeb flows: Katok constructed a metric with exactly two closed geodesics, while the corresponding general assertion is described in the source as a longstanding conjecture.