Miasnikov–Ventura–Weil algebraic-extension conjecture for free groups

Let $F_Y$ be a free group on the set $Y$, and let $H\leq K\leq F_Y$ be subgroups. For a basis $X$ of $F_Y$, write $\Gamma_X(H)\to\Gamma_X(K)$ for the morphism between the associated core graphs. An extension $H\leq K$ is algebraic if it is not contained in any proper free factor of $K$.

Miasnikov–Ventura–Weil conjecture. If the morphism

$$\Gamma_X(H)\to\Gamma_X(K)$$

is surjective for every basis $X$ of $F_Y$, then $K$ is an algebraic extension of $H$.

The converse is true, but the conjecture itself is refuted by a counterexample in the free group $F_2$ on two generators.