The Benjamini-Nekrashevych-Pete conjecture on strongly scale-invariant groups

Let $G$ be a finitely generated group. A group is strongly scale-invariant if it admits an injective homomorphism $f:G\to G$ such that $f(G)$ is a proper finite-index subgroup of $G$ and

$$\bigcap_{k\geq 0} f^k(G)$$

is finite. Benjamini-Nekrashevych-Pete conjecture. If $G$ is strongly scale-invariant, then $G$ is virtually nilpotent. The conjecture is known for virtually polycyclic groups, and this paper proves it for polycyclic groups of the form $\mathbb{Z}^n\rtimes\mathbb{Z}$; the general case remains open.