Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some base point. For each $\gamma\in \pi_1(\Sigma)$, there is a unique closed geodesic on $\Sigma$ in the free homotopy class of $\gamma$; let the length of this geodesic be $\ell(\gamma)$, which depends on the complex structure of $\Sigma$, of course.

Question: does there exist an $\epsilon =\epsilon(g) >0$, independent of the complex structure of $\Sigma$, such that for any $N\geq 2$ and elements $\gamma_1, \cdots, \gamma_N$ generating $ \pi_1(\Sigma)$, there is a $k$ with $\ell(\gamma_k) \geq \epsilon$?

Note that for each fixed $\Sigma$, and $\gamma$ non-trivial, $\ell(\gamma)$ is bounded below by the systole, so the question is really about a possible uniform bound over all choices of $\Sigma$. Any references or pointers would also be much appreciated!