The context-free quasigeodesic characterization of hyperbolic groups

Let $G$ be a finitely generated group. For rational $\lambda\geq 1$, real $\epsilon\geq 0$, and a finite generating set $S$, consider the language of $(\lambda,\epsilon)$-quasigeodesics in the Cayley graph $\Gamma(G,S)$. Say that $G$ has the $\mathbb{Q}\mathrm{CF}$ property when all these languages are context-free. Context-free quasigeodesic conjecture. A finitely generated group is hyperbolic if and only if it has the $\mathbb{Q}\mathrm{CF}$ property. This would extend the characterization of hyperbolic groups by regular quasigeodesic languages; the paper proves failure of the property for groups containing suitable undistorted nilpotent or Baumslag–Solitar subgroups, but the general converse remains open.