Uniformly almost flat quotients and virtual nilpotence

Let $G$ be a finitely generated residually finite group. For $d e 0$, write uniformly $(1/d)$-almost flat quotients and uniformly $(1/d)$-almost flat coset spaces for the corresponding properties defined in the paper. The uniformly almost flat quotients and virtual nilpotence conjecture. The group $G$ has uniformly almost flat quotients if and only if it has uniformly almost flat coset spaces, if and only if it is virtually nilpotent. Moreover, if $G$ has uniformly $(1/d)$-almost flat quotients, then $G$ has a finite-index nilpotent subgroup with Hirsch length at most $d$. The assertion is presented as a conjectural strengthening of the preceding claim. Its resolution is not supplied in the given text.