The finite abelian hierarchy conjecture for the $L^2$-Hall property

Let $G$ be a group admitting a finite abelian hierarchy. Suppose that $G$ is torsion-free and hyperbolic relative to virtually abelian subgroups.

Finite abelian hierarchy conjecture. Then $G$ should have the $L^2$-Hall property and satisfy the Geometric Hanna Neumann conjecture.

The paper presents this as a proposed class of groups satisfying the GHNC and as a unifying framework for the groups considered in its results. The supplied text gives no resolution of this statement.