Nekrashevych–Pete's conjecture on strongly scale-invariant groups

A finitely generated group $\Gamma$ is strongly scale-invariant if there exists an embedding $\varphi:\Gamma\hookrightarrow\Gamma$ with image of finite index and such that

$$\bigcap_n \varphi^n(\Gamma)$$

is finite. Nekrashevych–Pete's conjecture. Any strongly scale-invariant group is virtually nilpotent. This variant strengthens the scale-invariance question by requiring the nested finite-index subgroups to arise as iterated images of a single self-embedding; the preceding examples of scale-invariant, non-virtually nilpotent groups are not strongly scale-invariant. The source presents this as a proposed conjecture, and no resolution is supplied here.