Lehnert's co-context-free group conjecture

A finitely generated group $G$ is co-context-free if its co-word problem is context-free. Thompson's group $V$ is the group of homeomorphisms represented by finite binary tree-pair diagrams. Lehnert's conjecture. A finitely generated group $G$ is co-context-free if and only if it embeds in Thompson group $V$.

This conjecture proposes an algebraic characterization of groups with context-free co-word problem in terms of embeddability into Thompson's group $V$. Its resolution is not supplied in the source, so its status is left open.