3 problems

Inert equaliser conjecture. If $S$ contains at least one injective homomorphism, then $\operatorname{Eq}(S)$ is inert in $F(\Sigma)$.

Wilton's inclusion/exclusion Hanna Neumann conjecture. For all such $H$ and $K$,

Let $H$ be a subgroup of a free group $F$. It is compressed if for every subgroup $G<F$ containing $H$, one has $\operatorname{rk}(H)\leq\operatorname{rk}(G)$, and it is inert if f…