The hyperbolic Ehrenpreis conjecture

Let $\epsilon>0$ and let $S_1,S_2$ be two closed hyperbolic surfaces. A finite-sheeted locally isometric cover of a surface is a finite-sheeted covering surface equipped with the pulled-back hyperbolic metric. The hyperbolic Ehrenpreis conjecture. There exist finite-sheeted locally isometric covers $\widetilde{S}_i$ of $S_i$ for $i=1,2$ such that there is a $(1+\epsilon)$-bi-Lipschitz homeomorphism

$$\widetilde{S}_1\cong\widetilde{S}_2.$$

The conjecture concerns the common geometry of finite covers of closed hyperbolic surfaces. It is attributed in the source to Ehrenpreis and Goldman; the source does not state its resolution status.