The unit-interval range conjecture for irrational parameters

Let $\alpha\in(0,1)$ be irrational, and let $f_\alpha$ be the function defined in the paper. The unit-interval range conjecture.

$$\operatorname{Range}(f_\alpha)=\{0,1\}.$$

This is presented as the assertion that every irrational parameter in $(0,1)$ satisfies the corresponding equality in the paper's earlier proposition. It remains conjectural.