The Ehrenpreis conjecture for closed Riemann surfaces

Let $S_1$ and $S_2$ be closed Riemann surfaces of the same genus, and let $\tilde S_i$ denote covers of $S_i$ for $i=1,2$. A cover is conformal when the covering map is conformal, and a homeomorphism is $(1+\epsilon)$-quasiconformal when its quasiconformal distortion is at most $1+\epsilon$. Ehrenpreis's conjecture. For every $\epsilon>0$, there exist finite-sheeted conformal covers $\tilde S_i$ of $S_i$ such that there is a $(1+\epsilon)$-quasiconformal homeomorphism between $\tilde S_1$ and $\tilde S_2$. The conjecture was proved for tori, while the higher-genus case is equivalent to its hyperbolic formulation and is the subject of the paper's weak-form results.