The free-extension characterization of algebraic extensions

Let $H\leq J$ be subgroups of a free group $\mathbf{F}$. Let $\mathbf{F}'$ be a free extension of $\mathbf{F}$, meaning that $\mathbf{F}\leq\mathbf{F}'$ and $\mathbf{F}'$ is free, and let $X$ be a basis of $\mathbf{F}'$. Write $H\leq_{{\scriptscriptstyle \overset{\twoheadrightarrow}{X}}}J$ when the canonical morphism from the Stallings core graph $\Gamma_X(H)$ to $\Gamma_X(J)$ is onto. Modified algebraic-extension conjecture. Then

$$H\leq_{alg}J\quad\text{if and only if}\quad H\leq_{{\scriptscriptstyle \overset{\twoheadrightarrow}{X}}}J$$

for every free extension $\mathbf{F}'$ of $\mathbf{F}$ and every basis $X$ of $\mathbf{F}'$. This proposed modification is presented after the original conjecture is shown to fail, and the source gives no resolution of it.