Automatic-sequence uniformity conjecture

Let $a(n)$ be a $2$-automatic sequence such that

$$\operatorname*{\mathbb{E}}_{n<N}a(qn+r)\longrightarrow 0\quad\text{as }N\longrightarrow\infty$$

for every $q\in\mathbb{N}$ and $r\in\mathbb{N}_0$. Then the automatic-sequence uniformity conjecture.

$$\lVert a\rVert_{U^s[N]}\longrightarrow 0\quad\text{as }N\longrightarrow\infty$$

for every $s\in\mathbb{N}$.

The conjecture proposes that vanishing averages on every affine subsequence force vanishing Gowers uniformity norms for all orders. It is motivated by the paper's main theorems and is stated especially strongly for sequences with symmetric kernel; its general status is not resolved in the supplied text.