KöMaL 2022 A825

Find all functions $f\colon \mathbb{Z}^+\to \mathbb{R}^+$ that satisfy $f(nk^2)=f(n)f^2(k)$ for all positive integers $n$ and $k$, furthermore $\lim\limits_{n\to \infty} \frac{f(n+1)}{f(n)}=1$.