The co-context-free group embedding conjecture for Thompson's group V

A finitely generated group is co-context-free if its word problem's complement is context-free. The co-context-free embedding conjecture. A finitely generated group is co-context-free if and only if it embeds into Thompson's group $V$. This conjecture proposes Thompson's group $V$ as a universal container for finitely generated co-context-free groups.