Are there any 2-adic obstructions that prevent infinite regenerative cycles in Collatz odd-step block trajectories?

We consider the odd-step block transitions of the Collatz mapping by collapsing intermediate even divisions: $$3n_i + 1 = 2^{k_i} n_{i+1}, \quad \text{where } k_i = v_2(3n_i + 1)$$

We model potential upward expansion chains ($1^r$, where $k_i = 1$ for $r$ consecutive steps) followed by compression blocks ($C$) as a regenerative transition: $$F_r(n) = \left(\frac{3}{2}\right)^r(n+1) - 1$$

To test whether such regenerative patterns can extend indefinitely, we define the preimage sets for $j$ repetitions: $$A_j = \{ n_0 \in \mathbb{N} : \text{realizes a specified regenerative pattern } j \text{ times} \}$$

Question: Can it be proven that the iterative pullback of 2-adic residue conditions for non-trivial regenerative patterns eventually encounters a finite obstruction (i.e., $A_j = \emptyset$ for some finite $j$, or that no infinite solution exists in $\mathbb{Z}_2$ mapping to $\mathbb{N}$), or what specific 2-adic analytic obstruction prevents such cycles from extending infinitely?