Ehrenpreis conjecture for finite hyperbolic covers

Let $\Sigma$ and $\Sigma'$ be closed Riemann surfaces of genus greater than $1$, regarded as Riemannian manifolds with metrics of curvature $-1$. The Ehrenpreis conjecture concerns finite-degree isometric covers of these surfaces. Ehrenpreis conjecture. For any $\epsilon>0$, there are finite-degree isometric covers $Z\rightarrow\Sigma$ and $Z'\rightarrow\Sigma'$ whose total spaces are $(1+\epsilon)$-quasiisometric. This is the geometric, asymptotic analogue of the fact that the universal covers are both isometric to $\mathbb H^2$. The paper states that an $L^1$ version has been solved, while this formulation is presented as the traditional Ehrenpreis conjecture.