Abstract We show that for a minimal system (X, T), the set of saturated points along cubes with respect to its maximal -step pro-nilfactor X_ has a full measure. As an application, it is shown that if a minimal system (X, T) has no non-trivial (k+1) -tuples with arbitrarily long finite IP-independence sets, then it has only at most k ergodic measures and is an almost k' to one extension of X_ for some k' k. In particular, for k=1, we prove that (X, T) is uniquely ergodic (even regular with respect to X_), which answers a conjecture stated by Dong et al Infinite-step nilsystems, independence and complexity. Ergod. Th. & Dynam. Sys. 33 (1) (2013), 118–143.
Qiu et al. (Tue,) studied this question.