Maybe it is worth to write my comments into an answer:

1. The question is equivalent to asking whether all elements of the form $n/3^k$ end up in the collatz cycle.
2. It is still an open question whether there is an integer $n$ where the collatz iterates go to infinity
3. It is still an open question whether there are other integral positive Collatz cycles
4. Iterating with starting value $n/3^k$ for some $k$ and $n\neq 0$, we always end up with an integer after finitely many steps. Each $x\mapsto 3x+3^k$ decreases the exponent of $3$ in the denominator by one. So the question is equivalent to asking whether the conjectures 2,3 hold.

Maybe it is worth to write my comments into an answer:

1. The question is equivalent to asking whether all elements of the form $n/3^k$ end up in the collatz cycle.
2. It is still an open question whether there is an integer $n$ where the collatz iterates go to infinity
3. It is still an open question whether there are other integral positive Collatz cycles
4. Iterating with starting value $n/3^k$ for some $k$ and $n\neq 0$, we always end up with an integer after finitely many steps. Each $x\mapsto 3x+1$ decreases the exponent of $3$ in the denominator by one. So the question is equivalent to asking whether the conjectures 2,3 hold.