$L^1$ Ehrenpreis conjecture for the mapping class group of a surface solenoid

Let $\widehat{\Sigma}$ be the algebraic universal cover of a closed surface $\Sigma$, let $\ell$ be a base leaf, and let ${\sf M}(\widehat{\Sigma},\ell)$ be its mapping class group acting on the Teichmüller space $\mathcal{T}(\widehat{\Sigma})$. Equip this space with the topology induced by the $L^1$ metric. $L^1$ Ehrenpreis conjecture. The mapping class group ${\sf M}(\widehat{\Sigma},\ell)$ acts with $L^1$-dense orbits on $\mathcal{T}(\widehat{\Sigma})$. This is the $L^1$ refinement of the solenoidal Ehrenpreis conjecture and is the principal statement studied in the paper. The surrounding text says that the paper proves an $L^1$ version, so this candidate is treated as solved.