Ehrenpreis conjecture for the mapping class group of a surface solenoid

Let $\widehat{\Sigma}$ be the algebraic universal cover of a closed surface $\Sigma$, with Teichmüller space $\mathcal{T}(\widehat{\Sigma})$. Let ${\sf M}(\widehat{\Sigma},\ell)$ be the mapping class group of homotopy classes of homeomorphisms fixing a base leaf $\ell$, identified with homotopy classes of lifts of correspondences between finite covers of $\Sigma$. Ehrenpreis conjecture. The group ${\sf M}(\widehat{\Sigma},\ell)$ acts on $\mathcal{T}(\widehat{\Sigma})$ with dense orbits. This is the genus-independent, or solenoidal, formulation of the Ehrenpreis conjecture. It is closely related to the finite-cover and conformal formulations above; the paper subsequently studies its $L^1$ version.