Colored Stallings graphs and Counterexamples to Stallings equalizer conjecture

The famous Stallings equalizer conjecture has remained open for more than 40 years, which states that, for any free group $F_n$ of rank $n\ge 2$, any free group $F$, and any two monomorphisms $g,h:F_n\to F,$ the equalizer $\text{Eq}(g,h)=\{w\in F_n\mid g(w)=h(w)\}$ satisfies $\text{rk}\text{Eq}(g,h)\le n\mathrm{.}$ The only known case is $n=2$, due to A. D. Logan in 2022. By introducing the notion of colored Stallings graphs, we show that for every integer $n\ge 2$ there exist monomorphisms $g,h:F_n⟶F_2$ such that$\text{rk}\text{Eq}(g,h)\ge 2n-2.$ This disproves Stallings equalizer conjecture for $n\ge 3$.