The algebraic-extension characterization via Stallings graph coverings

Let $\mathbf{F}$ be a free group, and let $H\leq J\leq\mathbf{F}$. For a basis $X$ 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. An extension $H\leq J$ is algebraic when $J$ is not contained in any proper free factor of $\mathbf{F}$ containing $H$. Algebraic-extension conjecture. If $H\leq J\leq\mathbf{F}$ and $H\leq_{{\scriptscriptstyle \overset{\twoheadrightarrow}{X}}}J$ for every basis $X$ of $\mathbf{F}$, then $J$ is an algebraic extension of $H$. The converse was known: every algebraic extension has the covering property for every basis. The conjecture was raised by Martino, Valiunas and Ventura, but the paper explains that the original conjecture fails; this candidate is therefore refuted.