The universal quasigeodesic obstruction conjecture

Let $G$ be a non-hyperbolic finitely generated group with generating set $S$, and let $\Gamma(G,S)$ be its Cayley graph. A word is a $(\lambda,\epsilon)$-quasigeodesic when it satisfies the corresponding quasigeodesic inequalities. Universal quasigeodesic obstruction conjecture. There exists a universal constant $\lambda_0$ such that, for every such $G$ and $S$, and for all $\lambda>\lambda_0$, the $(\lambda_0,0)$-quasigeodesics in $\Gamma(G,S)$ do not form a context-free language. This is stronger than the paper's $\mathbb{Q}\mathrm{CF}$ conjecture because it seeks a uniform threshold; the techniques developed establish no such universal constant, so the conjecture remains open.