Conjecture on finite-to-one maximal equicontinuous factors of minimal actions

Let $G$ be an amenable group, ${G \curvearrowright X}$ a minimal action, and let $k\geq 2$. Write ${\rm IT}_k(X,G)$ for the set of independence tuples of length $k$, let $\Delta^2_k(X)$ denote the corresponding regionally proximal relation of order two, and let $\pi_{eq}:X\to X_{eq}$ be the maximal equicontinuous factor map. Finite-to-one factor conjecture. If

$${\rm IT}_k(X,G)\subset \Delta^2_k(X),$$

then $\pi_{eq}$ is almost finite-to-one. This concerns when restrictions on independence tuples force the maximal equicontinuous factor to have finite fibres; the conjecture was subsequently related to a result proved by Maass and Shao, so it is recorded as solved.