Canonical lift volume-minimizing conjecture for filling closed geodesics

Let $\Sigma$ be a hyperbolic surface, let $\gamma$ be a filling closed geodesic on $\Sigma$, let $\hat{\gamma}$ denote its canonical lift, and let $M_{\tilde{\gamma}}$ be the complement associated with a lift $\tilde{\gamma}\to\gamma$. Canonical lift volume-minimizing conjecture.

$$\operatorname{Vol}(M_{\hat{\gamma}})=\min_{\tilde{\gamma}\to\gamma}\{\operatorname{Vol}(M_{\tilde{\gamma}})\}.$$

The conjecture asks whether the canonical lift complement has the smallest volume among all lift complements of a fixed filling closed geodesic. Continuous lift complements are hyperbolic $3$-manifolds of finite volume, but the minimization assertion is left as a question.