The induction conjecture for uniformly almost flat quotients

Let $d e 0$, let $G$ be a finitely generated infinite residually finite group with uniformly $(1/d)$-almost flat quotients, and let $H$ be a finite-index subgroup admitting a quotient $H/K\cong\mathbb{Z}$. The induction conjecture for uniformly almost flat quotients. The subgroup $K$ is finitely generated and has uniformly $(1/(d-1))$-almost flat quotients. This conjecture is proposed as the inductive step that could reduce the general statement to the polycyclic case, analogously to reductions used in proofs of results on polynomial growth. The supplied text gives no resolution of this conjecture.