Polynomial Sarnak's conjecture for minimal systems III

Let $p:\mathbb{N}\to\mathbb{N}_0$ be a polynomial and let $(X,T)$ be a minimal topological dynamical system. For $f\in C(X)$ and $x\in X$, consider the sequence $(f(T^{p(n)}x))$. Polynomial Sarnak's conjecture for minimal systems III. If this sequence has topological entropy zero, then

$$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N\mu(n)f(T^{p(n)}x)=0.$$

This formulation imposes the entropy-zero condition directly on the sampled observable sequence rather than on the whole system. The source presents it as a conjecture and gives no resolution; minimality remains part of the hypothesis.