20.11 (2022)

Let $F \leqslant H$ be free groups such that there exists a free group $G$ of finite rank with $F \leqslant G \leqslant H$, and let $r$ be the least of the ranks of such groups $G$. Which, if any, of the following statements must hold?
$\qquad$ (i) There is a largest $G$ of rank $r$ between $F$ and $H$.
$\qquad$ (i$'\,$) For any two $G_1$, $G_2$ of rank $r$ between $F$ and $H$, the subgroup $\langle G_1, G_2 \rangle$ has rank $r$.
$\qquad$ (ii) There is a smallest $G$ of rank $r$ between $F$ and $H$.
$\qquad$ (ii$'\,$) For any two $G_1, G_2$ of rank $r$ between $F$ and $H$, the subgroup $G_1 \cap G_2$ has rank $r$.

If (i) holds for all such $F$ and $H$, then so does (i$'\,$). If (ii) holds for all $F, H$, then so does (ii$'\,$). The converse of the former implication holds because subgroups of a free group of any fixed finite rank satisfy ACC, but I don’t see a way to get the converse of the other implication.