Infinite equalizer conjecture for free-group endomorphisms

Let $F_n$ be a finitely generated free group, and let $\phi,\psi$ be injective endomorphisms of $F_n$. Their extensions to the boundary are denoted by $\widehat\phi$ and $\widehat\psi$, and write $\operatorname{Eq}(\phi,\psi)$ and $\operatorname{Eq}(\widehat\phi,\widehat\psi)$ for the corresponding equalizers. The regular points of $\operatorname{Eq}(\widehat\phi,\widehat\psi)$ are the points considered in the paper's boundary equalizer construction.

Infinite equalizer conjecture. There is an upper bound, depending only on $n$, for the number of $\operatorname{Eq}(\phi,\psi)$-orbits of the regular points of $\operatorname{Eq}(\widehat\phi,\widehat\psi)$. The finite-rank-two case of the related equalizer problem is known, but the general case remains open; this infinite version is posed as an open problem.