The generalized Collatz orbit criterion

Let $m\in\mathbb{N}$, and let $\operatorname{Col}$ be the Collatz function defined by

$$\operatorname{Col}(x)=\begin{cases}\frac{x}{2}&\text{if }x\equiv0\pmod 2,\\3x+1&\text{if }x\equiv1\pmod 2.\end{cases}$$

Consider the sequence $\operatorname{Col}^n(m)$ for $n\geq1$. Generalized Collatz conjecture. This sequence will reach $1$ if and only if both of the following properties hold:

1. $\displaystyle\lim_{n\to\infty}\operatorname{Col}^n(m)\notin\{m\}$.
2. $\displaystyle\lim_{n\to\infty}\operatorname{Col}^n(m)\neq\infty$.

This is presented as a reformulation of the Collatz conjecture in the source. The limiting conditions are not further clarified there, and no resolution or independent justification of the claimed equivalence is supplied.