Let $G_1 \to G_2 \to \cdots$ be a sequence of epimorphisms of finitely generated residually finite groups. Does it eventually stabilize?

The analogous result holds when the groups are residually free: this is Proposition 6.8 in Charpentier Guirardel ["Limit groups as limits of free groups"](https://eclass.uoa.gr/modules/document/file.php/MATH154/%CE%88%CE%B3%CE%B3%CF%81%CE%B1%CF%86%CE%B1%20%CE%A0%CE%B1%CE%BB%CE%B1%CE%B9%CF%8E%CE%BD%20%CE%A3%CE%B5%CE%BC%CE%B9%CE%BD%CE%B1%CF%81%CE%AF%CF%89%CE%BD/Guirardel-Champetier.pdf). The proof only uses the fact that residually free groups are residually $SL_2(\mathbb{C})$, and it seems that it can be adapted to the case where each $G_i$ is residually $GL_n(\mathbb{C})$ for a fixed $n$. It seems unlikely that this holds for general residaully finite groups: the [Jordan-Schur Theorem](https://groupprops.subwiki.org/wiki/Jordan-Schur_theorem_on_abelian_normal_subgroups_of_small_index) implies that for a general finite group the minimal degree $n$ such that it embeds into $GL_n(\mathbb{C})$ can be arbitrarily large.

Is there another way to adapt the proof? Is there a counterexample?