The Schreier Girth Alternative for linear groups

Let $G$ be a finitely generated group, and define its Schreier girth by

$$\mathcal{S}\text{-girth}(G)=\inf_{\langle S\rangle=G}\left\{\sup_{\langle S'\rangle=G,\,S'\sim_{\mathcal{S}} S}\operatorname{girth Cay}(G,S')\right\},$$

where $S$ is a minimal generating set and $S\sim_{\mathcal{S}}S'$ means that $S'$ can be obtained from $S$ by finitely many Schreier transformations. Schreier Girth Alternative. Any finitely generated linear group is either virtually solvable or has an infinite $\mathcal{S}$-girth. The source identifies this as an open problem and describes it as one of the paper's major motivations.