Inert equaliser conjecture for free-group homomorphisms

Let $F(\Sigma)$ and $F(\Delta)$ be free groups, and let $S:F(\Sigma)\rightarrow F(\Delta)$ be a set of homomorphisms. Define

$$\operatorname{Eq}(S):=\bigcap_{g,h\in S}\operatorname{Eq}(g,h),$$

where $\operatorname{Eq}(g,h):=\{x\in F(\Sigma)\mid g(x)=h(x)\}$. A subgroup $H$ of a free group $F(\Sigma)$ is inert if, for every subgroup $K\leq F(\Sigma)$,

$$\operatorname{rk}(H\cap K)\leq \operatorname{rk}(K).$$

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

This conjecture generalises the Dicks--Ventura result that fixed subgroups of monomorphisms are inert. The paper uses inertness to study equalisers and proves the relevant rank-two result, but the conjecture as stated is not identified as resolved in the supplied text.