Parzanchevski–Puder revised algebraic-extension conjecture for free groups

Let $F_Y$ be a free group with subgroup chain $H\leq K\leq F_Y$. Let $F'$ be a free extension of $F_Y$, and let $X$ be a basis of $F'$. The associated core graphs give a morphism $\Gamma_X(H)\to\Gamma_X(K)$. An extension $H\leq K$ is algebraic if it is not contained in any proper free factor of $K$.

Parzanchevski–Puder conjecture. If, for every free extension $F'$ of $F_Y$ and every basis $X$ of $F'$, the morphism

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

is onto, then $K$ is algebraic over $H$.

This was proposed as a revision of the refuted Miasnikov–Ventura–Weil conjecture; the supplied text does not establish whether this revised conjecture is resolved.