Subbarao–Fabrykowski conjecture on quasi-multiplicative functions

Let $f:\mathbb{N}\longrightarrow\mathbb{Z}$ be quasi-multiplicative, meaning that $f(pn)=f(p)f(n)$ whenever $p$ is prime and $p\nmid n$. Define

$$T_f=\left\{p\text{ prime}:\forall n\in\mathbb{N},\ f(n+p)\equiv f(n)\pmod p\right\}.$$

Subbarao–Fabrykowski conjecture. If $T_f$ is infinite, then either $f(n)=0$ for every $n\in\mathbb{N}$ or $f$ is a global power map, meaning $f(n)=n^k$ for some $k\in\mathbb{N}\cup\{0\}$. The paper cites this as a conjecture of Subbarao and Fabrykowski and discusses progress toward it, including a positive-upper-density result.