Existence of closed geodesics on certain non-compact Riemannian manifolds

Let $M$ be a complete Riemannian manifold. Suppose $M$ contains a bounded, concave, connected open set $U$ with $C^0$ boundary and $M\setminus U$ is connected. We assume that either the relative homotopy set $π_1(M,M\setminus U)=0$ or the union of all the conjugate subgroups of the image of the homomorphism $π_1(M\setminus U)\rightarrow π_1(M)$ (induced by the inclusion $M\setminus U\hookrightarrow M$) is a proper subset of $π_1(M)$. (The first condition is equivalent to $π_1(M\setminus U)\rightarrow π_1(M)$ is surjective; the second condition is satisfied if the relative homology group $H_1(M,M\setminus U)\neq 0$.) Then there exists a non-trivial closed geodesic on $M$. This partially proves a conjecture of Chambers, Liokumovich, Nabutovsky and Rotman.