The range conjecture for supra-unitary rational parameters

For $b\in\mathbb{N}\setminus\{1\}$, let $a\in\{1,\ldots,b-1\}$ be relatively prime to $b$, and let $s\in\mathbb{N}$. For the function $f_\alpha$ appearing in the paper, the range conjecture asserts

$$\operatorname{Range}\bigl(f_{sb-\frac{a}{b}}\bigr)=\bigl(s\{0,\ldots,b-1\}\bigr)\cup\bigl(s\{1,\ldots,b-1\}-\{1\}\bigr).$$

The authors report computational evidence for this exact description but do not know how to prove it in general; only various general cases and small-denominator cases have been settled.