The golden-ratio characterization by constant range

Let $\alpha>0$ be irrational, let $f_\alpha$ be the function defined in the paper, and let $\varphi$ denote the golden ratio. The golden-ratio range conjecture.

$$\operatorname{Range}(f_\alpha)=\{1\}\qquad\Longleftrightarrow\qquad\alpha=\varphi.$$

The authors state that the golden ratio is the only positive irrational parameter for which they encountered a constant function in their explorations. The equivalence is conjectural and unproved.