A consequence of the famous Jørgensen inequality is that there is a lower bound for the distance between closed geodesics in hyperbolic three-manifolds: for any $R>0$ there is a c>0 such that for any such manifold $N$ and any two distinct closed geodesics loops $c_1,c_2$ on $M$ both of length less than $R$ we have $$ d_H(c_1,c_2) \ge c \quad (\ast) $$ where $d_H$ is Hausdorff distance. This is false for general locally symmetric spaces, since the higher-rank ones can contain flat tori. My question is the following: if we fix a real-rank one symmetric space $X$ (i.e. $X$ is a real, complex or quaternionic hyperbolic space, or octonionic hyperbolic plane) is there a c>0 (depending on the space) such that $(\ast)$ holds for all quotients of $X$ and all pairs of distinct closed geodesics of bounded length on such? (More generally one could ask the same question for Riemannian $n$-manifolds with sectional curvature between $-1$ and $\kappa<0$, with a $c$ depending on $\kappa$ and $n$ and the bound on the length).
Edit: As noted in Anton's comments the opening remark is false if one does not bound the length of the geodesics; I edited to add that.