Effective virtual Haken conjecture

Let $N$ be a closed hyperbolic 3-manifold. A finite-sheeted Haken covering of $N$ is a finite-sheeted covering space of $N$ that is Haken. Effective virtual Haken conjecture. Every closed hyperbolic manifold $N$ has a finite-sheeted Haken covering, with an $\mathsf{ER}$ bound on the number of sheets. This is an effective-complexity version of the virtual Haken theorem, which is known qualitatively but whose placement in $\mathsf{ER}$ is presented here as open.