Around first-order rigidity of Coxeter groups

By the work of Sela, for any free group $F$, the Coxeter group $W_3 = \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$ is elementarily equivalent to $W_3 \ast F$, and so Coxeter groups are not closed under elementary equivalence among finitely generated groups. We stud