Finsler uniformization conjecture for reversible metrics on the projective plane

Let $L$ be a reversible Finsler metric on ${\mathbb R}P^2$. A smooth positive function $\rho$ on ${\mathbb R}P^2$ is a function whose product with $L$ defines the conformally rescaled Finsler metric $\rho L$. Finsler uniformization conjecture. There exists a smooth positive function $\rho$ on ${\mathbb R}P^2$ such that the geodesic flow of the metric $\rho L$ is periodic. This would give a Finsler generalization of the uniformization theorem and, together with the preceding isosystolic inequality, would imply the Finsler extension of Pu's theorem for the projective plane.