open [@BM19 Question 2.3.2].

In order to put these results in perspective, we recall some known results concerning existence of closed geodesics on complete non-compact Riemannian manifolds. In 1978, Thorbergsson proved the existence of closed geodesics on a complete Riemannian manifold $M$ if it contains a convex compact set which is not homotopically trivial or if $M$ has a non-negative sectional curvature outside some compact set [@Tho78]. In the 1990s, Benci and Giannoni proved that any complete $d$-dimensional Riemannian manifold such that the limit superior of its sectional curvature at infinity is non-positive and the homology of its free loop space is non-trivial in some degree larger than $2d$ possesses a closed geodesic [@BG91; @BG92]. In 2017, Asselle and Mazzucchelli showed the existence of infinitely many closed geodesics for complete $d$-dimensional Riemannian manifolds which have no close conjugate points at infinity and a free loop space with unbounded Betti numbers in degrees larger than $d$ [@AM17]. They also improved the result of Benci and Giannoni by replacing the asymptotic curvature assumption by an assumption on the conjugated points at infinity and by improving the bound on the homology of the free loop space. However, the existence of one closed geodesic 