Coboundary characterization of nonfluctuating irrational rotation averages

Let $\theta otin\boldsymbol{Q}$ and let $4\mathsf{x}={x_n}_n$ with $x_n=n\theta$. For a continuous function $f$, write

$$\mathbb{A}_{[N]}f(\mathsf{x})=\frac{1}{N}\sum_{n\in[N]}f(\{n\theta\}).$$

We say that these averages fluctuate when they are infinitely often greater than and infinitely often less than their limiting value; here the relevant limiting value is $0$. Coboundary characterization conjecture. If, for a continuous function $f$, the averages $\mathbb{A}_{[N]}f(\mathsf{x})$ fail to fluctuate on a set of positive measure, then there is a real-valued, non-constant function $F$ and a fixed real number $\theta$ such that

$$f(t)=F(t)-F(\{t+\theta\}).$$

The claim proposes a necessary condition for failure of fluctuation in irrational rotation averages; the source leaves the corresponding necessary-and-sufficient characterization unresolved.