The conjecture on non-contractible closed geodesics on the Riemannian real projective plane

Let $coperatorname{\mathbb{R}P}^2$ be a real projective plane endowed with a Riemannian metric $g$. A closed geodesic is non-contractible if its underlying loop represents a nontrivial element of the fundamental group of $coperatorname{\mathbb{R}P}^2$. Two prime closed geodesics are distinct if they are not related by a time translation. Non-contractible closed-geodesic growth conjecture. There exist infinitely many distinct non-contractible closed geodesics on $(coperatorname{\mathbb{R}P}^2,g)$. Moreover, the number of non-contractible closed geodesics of length at most $l$ grows at least like the prime numbers. Motivated by known results under positive Gaussian curvature, this conjecture proposes both infinitude and a prime-number lower growth rate without that curvature assumption; the source gives no resolution status.