Stallings' equaliser conjecture for free groups

Let $F(\Sigma)$ and $F(\Delta)$ be free groups, and let $g,h:F(\Sigma)\rightarrow F(\Delta)$ be homomorphisms. Write

$$\operatorname{Eq}(g,h):=\{x\in F(\Sigma)\mid g(x)=h(x)\}.$$

Here $\operatorname{rk}(H)$ denotes the rank of a free group $H$.

Stallings' equaliser conjecture. If $g$ and $h$ are homomorphisms with $h$ injective, then

$$\operatorname{rk}(\operatorname{Eq}(g,h))\leq |\Sigma|.$$

This conjecture concerns the finite generation and rank of equalisers of free-group homomorphisms when one map is injective. The source attributes it to Stallings; the paper proves a strong form for the free group of rank two, while the stated general bound is not presented as resolved here.