The Collatz orbit conjecture

Let

$$f(n)=\begin{cases}\frac{3n+1}{2},&\text{if }n\text{ is odd},\\frac{n}{2},&\text{if }n\text{ is even.}\end{cases}$$

Let ${\cal O}_1=\{1,2\}$. An eventual finite orbit is a finite orbit in which the iterates of an initial integer eventually remain. Collatz orbit conjecture. ${\cal O}_1$ is the only eventual finite orbit that the Collatz function admits.

This is the uniqueness-of-the-eventual-orbit formulation of the Collatz conjecture. The source presents it as the goal of improving results concerning uniqueness, but provides no resolution of the claim.