Huang–Lian–Shao–Ye conjecture on almost finite-to-one maximal equicontinuous factors

Let $(X,G)$ be a minimal topological dynamical system, where $G$ is an amenable group, and suppose that $(X,G)$ admits no essential $K$-IT-tuples. Let $\pi_{eq}:X\to X_{eq}$ be the factor map to the maximal equicontinuous factor. Huang–Lian–Shao–Ye's conjecture. Then $\pi_{eq}$ is almost finite to one, meaning that there exists a finite $K$ such that

$$\{y\in X_{eq}:|\pi_{eq}^{-1}(y)|=K\}$$

is a residual subset of $X_{eq}$. This conjecture concerns the relationship between the absence of essential independence tuples and the fiber structure of the maximal equicontinuous factor map; its status is not resolved in the supplied source.