The finitely generated groups twin-width conjecture

Let $\Gamma$ be a countable group generated by a finite set $S$, and let $\operatorname{Cay}(\Gamma,S)$ be its Cayley graph with vertex set $\Gamma$ and edges $\{x,x\cdot s\}$ for $x\in\Gamma$ and $s\in S$. Let $F(\Gamma,S)$ be the class of all finite induced subgraphs of $\operatorname{Cay}(\Gamma,S)$.

Finitely generated groups conjecture. For every group $\Gamma$ generated by a finite set $S$, the class $F(\Gamma,S)$ has bounded twin-width.

The question is motivated by the examples of free groups and the integer grid, whose associated classes have bounded twin-width. The supplied text presents this as the main question of the section and gives no resolution.