Let $z$ be a point in a fundamental domain of $\Gamma(2)\subset \mathrm{SL}_2(\mathbf{Z})$ in the complex upper half plane.
Does there exist an $\epsilon >0$ such that the geodesic distance $d(z,\gamma z)$ satisfies
$ \cosh( d(z,\gamma z)) > 1+\epsilon$ for all $\gamma \neq \pm 1$ in $\Gamma(2)$?
Note that $\epsilon$ will probably depend on $z$.