Is there a Collatz-like sequence ending with $1,1,1,\ldots$?

Question

I found a conjecture of some mathematics teacher. Is this known or solved problem? Is there a non-constant function $f$ such that for any $n$, iterating $f(n)$ end to the sequence $1,1,1,1,1,\ldots$. By iteration I mean the similar way as in Collatz problem we end to the sequence $2,1,2,1,\ldots$. For example, does the function $f(n)=\begin{cases}n/3\text{ if }n\equiv 0\pmod 3 \\(4n-1)/3\text{ if }n\equiv 1\pmod 3\\(5n-7)/3\text{ if} n\equiv 2\pmod 3\end{cases}$ end to the sequence $\ldots, 1,1,1,\ldots$ This is from the Finnish mathematical journal from college students called Solmu, http://matematiikkalehtisolmu.fi/2017/1/otaksuma.pdf