The divergent trajectories conjecture for the Collatz map

Let $C:\mathbb{Z}\to\mathbb{Z}$ be the Collatz map and let $\mathbb{N}_{1}$ denote the positive integers. A trajectory is divergent when its iterates are unbounded. Divergent trajectories conjecture. The map $C$ has no divergent trajectories in $\mathbb{N}_{1}$; equivalently, for every $n\in\mathbb{N}_{1}$,

$$\sup_{k\geq0}C^{\circ k}(n)<\infty.$$

Together with the assertion about positive periodic points, this is described as the second component of the Collatz conjecture and remains open.