Sharp rank-$2n-2$ equalizer bound for monomorphisms

Let $n\ge 3$, let $F_n$ be a free group of rank $n$, and let $F$ be a free group. Let $g,h:F_n\to F$ be monomorphisms. Sharp equalizer rank conjecture. Then

$$\operatorname{rk}\operatorname{Eq}(g,h)\le 2n-2.$$

The paper constructs examples attaining rank at least $2n-2$, motivating this as a candidate for the correct sharp upper bound after the Stallings equalizer conjecture was disproved. Its status is open in the supplied text.