Let $M$ be a smooth compact Riemannian manifold. Let $\varepsilon>0$. I call a smooth curve $\gamma\colon [a,b]\to M$ an $\varepsilon$-geodesic if there is $\delta>0$ such that for any $t_1<t_2<t_1+\delta$ one has $$dist(\gamma(t_1),\gamma(t_2))\leq length(\gamma[t_1,t_2])\leq (1+\varepsilon)(\gamma(t_1),\gamma(t_2)),$$ where $dist$ denotes the Riemannian distance between the two points, and $length(\gamma[t_1,t_2])$ is the length of the curve $\gamma$ between $t_1$ and $t_2$.

Question. Given $M$, do there exist $\varepsilon>0, l>0$ such that no $\varepsilon$-geodesic of length less than $l$ is a loop, i.e. $\gamma(a)\ne \gamma(b)$?

Remark. Clearly $0$-geodesics are usual geodesics. If the length of a geodesic is less then the injectivity radius then it cannot be a loop.

This question is a more precise version of "Almost geodesics" in Riemannian manifolds which cannot be loops