Huang–Lian–Shao–Ye conjecture on essential IT-tuples for proximal factors

Let $(X,G)$ be a minimal topological dynamical system, where $G$ is an amenable group, and let $\pi_{eq}:X\to X_{eq}$ be the factor map to the maximal equicontinuous factor. Suppose that $\pi_{eq}$ is proximal and not almost one to one. Huang–Lian–Shao–Ye's conjecture. For every integer $K\geq 2$, there exists an essential $K$-IT-tuple. This conjecture predicts that a proximal maximal equicontinuous factor map with nontrivial generic fibers forces essential independence tuples of every finite length; its status is not resolved in the supplied source.