Behrstock–Neumann's cusp covering conjecture

Let $M$ be a hyperbolic $n$-manifold, and let $C$ range over its cusps. For each cusp $C$, let $\pi_1(C)$ be its cusp fundamental group. A sublattice $\Lambda'_C\subseteq\Lambda_C$ determines a cover of $C$.

Cusp covering conjecture. For each cusp $C$ of $M$, there exists a sublattice $\Lambda_C$ of $\pi_1(C)$ such that, for every choice of a sublattice $\Lambda'_C\subseteq\Lambda_C$ for each cusp $C$, there is a finite regular cover $M'$ of $M$ whose cusps covering each cusp $C$ are precisely the covers determined by $\Lambda'_C$.

This conjecture concerns simultaneously prescribing cusp covers in a finite regular cover. The source presents it as a stronger version of the Behrstock–Neumann formulation and relates it to constructing compatible normal subgroups in higher-dimensional graph manifolds; its resolution is not supplied here.