The cycle-vertex conjecture for the Loosened Collatz Graph

Let $\mathbb{N}_1$ denote the positive integers, and let the Loosened Collatz Graph be the graph whose vertices and cycles are defined by the Loosened Collatz Function. Cycle-vertex conjecture. For all $n \neq 0 \text{ (mod 3)} \in \mathbb{N}_1$, $n$ is a vertex of a cycle. This conjecture asserts that every positive integer not divisible by $3$ lies on a cycle in the graph; the source gives no resolution, so its status remains open.