Does there exist a finitely generated non-finitely presentable group $$ G=\langle S \mid R\rangle $$ with $S$ finite, where we can enumerate $R=\{r_1,r_2,\ldots\}$ such that for every finite subset $I\subset \mathbb{N}$, the group $$ G_I=\langle S \mid \{r_i\mid i\in I\}\rangle $$ is virtually free?

One potential candidate is the lamplighter $\mathbb{Z}_2\wr\mathbb{Z}$ with the presentation $$ \langle a,t \mid a^2, [a,t^kat^{-k}] \textrm{ for all } k\in\mathbb{Z}\rangle. $$

(edit) Let me ask a more specific question related to lamplighters. Is it true that if $$ G_I=\langle a,t \mid a^2, [a,t^kat^{-k}] \textrm{ for all } k\in I\rangle $$ is virtually free for some finite set $I\subset\mathbb N$, then for every large enough $k$, $G_{I\cup\{k\}}$ is also virtually free?

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