Hyperbolicity of links complements in Seifert fibered spaces

Let $\bar\gamma$ be a link in a Seifert fibered space $M$ over a hyperbolic $2$-orbifolds $\mathcal O$ that projects injectively to a filling multicurve of closed geodesics $\gamma$ in $\mathcal O.$ We prove that the complement $M_{\bar\gamma}$ of $\bar\gamma$ in $M$ admits a hyperbolic structure of finite volume and give combinatorial bounds of its volume.