Bounded-deviation strengthening for closed tracking geodesics

Let $f\in\operatorname{Homeo}_0(S)$, where $S$ is a closed surface of genus $g\ge 2$. Suppose that the set of contractible fixed points of $f$ is inessential. Let $\gamma$ be a closed geodesic that is a tracking geodesic for some $\mu\in\mathcal{M}^{\mathrm{erg}}_{\vartheta>0}(f)$. For a lift $\widetilde\gamma$ of $\gamma$, define

$$V_C(\widetilde\gamma)=\{\widetilde x\in\widetilde S\mid d(\widetilde x,\widetilde\gamma)<R\}.$$

Bounded-deviation strengthening. There exists $C>0$ such that if an orbit $\widetilde y,\dots,\widetilde f^{n_0}(\widetilde y)$ of $\widetilde f$ crosses $V_C(\widetilde\gamma)$ for some lift $\widetilde\gamma$ of $\gamma$, then there exists an $f$-periodic orbit with one lift to $\widetilde S$ having its tracking geodesic intersecting $\widetilde\gamma$.

This strengthens the preceding bounded-deviations theorem by replacing crossings of many different lifts with a crossing of a prescribed neighborhood of one lift. The conjecture is presented under the additional inessentiality assumption on contractible fixed points; no resolution is supplied in the source.