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". 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 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?