  We show that for a minimal system $(X,T)$, the set of saturated points along
cubes with respect to its maximal $\infty$-step pro-nilfactor $X_\infty$ 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 $X_\infty$ for some $k'\leqslant k$. Particularly, for $k=1$
we prove that $(X,T)$ is uniquely ergodic (even regular with respect to
$X_\infty$), which answers a conjecture stated in [3].
