Let $\mathbb{N}$ denote the set of positive integers. The *Collatz function* $f:\mathbb{N}\to\mathbb{N}$ is given by $f(n) = n/2$ for $n$ even and $f(n) = 3n+1$ for $n$ odd. Given $k\in\mathbb{N}$ we associate to $k$ its *Collatz sequence* $(c^{(k)}_n)_{n\in\mathbb{N}}$ given inductively by $$c^{(k)}(1) = k\text{ and } c^{(k)}_{n+1} = f(c^{(k)}_n)\text{ for all } n\geq 1.$$
One version of the Collatz conjecture states that $$1\in \text{im}(c^{(k)}) \text{ for all }k\in\mathbb{N}.$$
Note that for all $k\in\mathbb{N}$ the sequence $c^{(k)}$ is either injective or eventually periodic. So any $c^{(k)}$ with $1\notin \text{im}(c^{(k)})$ would be either

(1) injective or

(2) eventually contain a period not containing $1$.

**Question.** Can the current state of research exclude one of the cases (1), (2)?