Wilton's inclusion/exclusion Hanna Neumann conjecture

Let $F$ be a finitely generated non-trivial free group, and let $H,K$ be finitely generated subgroups of $F$. Define the reduced rank of $H$ by

$$\overline r(H)=\max(0,\operatorname{rank}(H)-1).$$

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

$$\overline r(H\vee K)\,\overline r(H\cap K)\leq \overline r(H)\,\overline r(K),$$

where $H\vee K$ is the subgroup generated by $H\cup K$. This is known to hold when $K$ has finite index in $H\vee K$, but the paper gives counterexamples without that assumption, so the conjecture is refuted in its stated generality.