Pilatte's eventual periodicity conjecture for binary multiplicative functions

Let $f:\mathbb{N}\to\{0,1\}$ be a function, let $E\subseteq\mathbb{T}$ have positive measure, and consider the exponential sums

$$\sum_{n\leq x}f(n)e(n\alpha).$$

Pilatte's conjecture. These sums are bounded uniformly for all $x>0$ and all $\alpha\in E$ if and only if $f$ is eventually periodic. The conjecture was communicated to the authors by Cédric Pilatte and is motivated by work of Fregoli. The paper proves it when $f$ is completely multiplicative, while the general case remains open.