Geodesics and geodesic circles in a geodesically convex surface: a sub-mixing property

Let $M$ be an orientable finitely connected and geodesically convex Finsler surface with genus $g\ge 1$. We prove that if all geodesics in $M$ are reversible, then for any number $\varepsilon>0$ and for any points $p,q\in M$, there exists a number $R>0$ such that any geodesic circle with center $p$ and radius $t$ meets the $\varepsilon$-ball with center $q$ for any $t>R$. Most of the proofs do not use the reversibility assumption for geodesics.