Conformal Ehrenpreis conjecture

Let $\Sigma$ be a fixed compact surface of genus greater than $1$, let $\mathcal{T}(\Sigma)$ be its Teichmüller space, and let $d_{\mathcal{T}(\Sigma)}$ be the Teichmüller metric. Given $\mu,\nu\in\mathcal{T}(\Sigma)$, finite covers $\rho,\sigma:Z\rightarrow\Sigma$ induce isometric inclusions $\rho^*,\sigma^*: \mathcal{T}(\Sigma)\hookrightarrow\mathcal{T}(Z)$. Conformal Ehrenpreis conjecture. For any $\epsilon>0$, there exists a surface $Z$ and finite covers $\rho,\sigma:Z\rightarrow\Sigma$ such that

$$d_{\mathcal{T}(Z)}(\rho^*\mu,\sigma^*\nu)<\epsilon.$$

This is the conformal or traditional Teichmüller-theoretic formulation of the Ehrenpreis conjecture. The source presents it as a conjectural version and notes that the analogous statement in genus $1$ is not difficult to verify.