Universality conjecture for Thompson's group V and coCF groups

Let $V$ be Thompson's group, and let $\mathcal{V}$ be the class of finitely generated subgroups of $V$. Let $co\mathcal{CF}$ be the class of groups with context-free co-word problem.

Universality conjecture for $V$. The classes $\mathcal{V}$ and $co\mathcal{CF}$ are equal:

$$\mathcal{V}=co\mathcal{CF}.$$

Equivalently, Thompson's group $V$ is a universal $co\mathcal{CF}$ group.

The conjecture is motivated by the similar closure properties of $\mathcal{V}$ and $co\mathcal{CF}$, together with the fact that $V$ itself has context-free co-word problem. The source gives no resolution.