Arithmetic oscillation disjointness conjecture for zero-entropy torus flows

Let ${\bf c}=(c_n)_{n\in\mathbb N}$ be a sequence of complex numbers. For every real-coefficient polynomial $P$ of degree at most $d$ and every pair of integers $0\leq l<k$, suppose

$$\lim_{N\to\infty}\frac{1}{N}\sum_{1\leq n=mk+l\leq N}c_n e^{2\pi iP(n)}=0,$$

with the control condition that there is a constant $C$ such that

$$\frac{1}{N}\sum_{n=1}^N|c_n|^\lambda\leq C,\qquad \forall N\geq1.$$

Such a sequence is oscillating of order $d$ in the arithmetic sense. Arithmetic oscillation disjointness conjecture. An oscillating sequence ${\bf c}$ of order $d$ for all $d\geq2$ in the arithmetic sense is linearly disjoint from every flow on a torus with zero topological entropy. The conjecture proposes that arithmetic cancellation against polynomial phases suffices for linear disjointness from zero-entropy torus dynamics. The source gives no evidence of a general resolution.