The weak Collatz conjecture on finite cycles

Let $C:\mathbb{Z}\to\mathbb{Z}$ be the Collatz map, and let $\mathbb{N}_{1}$ denote the positive integers. A periodic point is an integer lying on a finite cycle of $C$. Weak Collatz conjecture. The only periodic points of $C$ in $\mathbb{N}_{1}$ are $1$, $2$, and $4$. This is also called the no finite cycles conjecture and is presented as the finite-cycle component of the Collatz conjecture; the source gives no resolution.