The local-global principle for power maps on the rational numbers

Let $f:\mathbb{Q}\longrightarrow\mathbb{Q}$. For each prime $p$, define

$$S_f=\left\{p\text{ prime}:\exists k_p\in\mathbb{Z}/(p-1)\mathbb{Z}\text{ such that }\forall \alpha\in\mathbb{Z}_{(p)}^\times,\ f(\alpha)\equiv\alpha^{k_p}\pmod p\right\},$$

where $\mathbb{Z}_{(p)}^\times$ is the group of elements of $\mathbb{Q}$ whose numerator and denominator are not divisible by $p$. Call $f$ a local power map at an infinite set of primes when $S_f$ is infinite, and a global power map when there is an integer $k\in\mathbb{Z}$ such that $f(\alpha)=\alpha^k$ for every $\alpha\in\mathbb{Q}$. The rational local-global conjecture for power maps. If $S_f$ is infinite, then $f$ is a global power map. The paper states that this conjecture implies the natural-number version; its resolution is not supplied in the text.