Quaternionic lattices and poly-context-free word problem

A finitely generated group $G$ is called poly-context-free if its word problem $\mathrm{W}\mathrm{P}(G)$ is an intersection of finitely many context-free languages. We consider the quaternionic lattices ${\mathrm{\Gamma }}_{\tau }$ over the field ${\mathbb{F}}_q(t)$ constructed by Stix-Vdovina (2017), and prove that they are not poly-context-free. As a corollary, since all the groups ${\mathrm{\Gamma }}_{\tau }$ are quasi-isometric to $F_2\times F_2$, the class of groups with poly-context-free word problem is not closed under quasi-isometries. The result follows from the description of the language $\mathrm{W}\mathrm{P}({\mathrm{\Gamma }}_{\tau })\cap a^{\ast }{b}^{\ast }{c}^{\ast }{d}^{\ast }$, which relies on the existence of anti-tori and certain power-type endomorphisms of the groups ${\mathrm{\Gamma }}_{\tau }$.