Does the limit $\lim_{n\to\infty} \frac{f(2n)}{f(n)}$ tend to infinity?

Kovač and Luca [KoLu25] (building on a heuristic independently found by Cambie (personal communication)) have shown that there is no finite limit, in that $\lim_{n\to\infty} \frac{f(2n)}{f(n)}$ is unbounded.