This question is somehow quite similar to the one posted here: How do I show convergence in the 2-adics? by "it's a hire car baby". We know that $\lim_{n\to\infty} p^n=0$ (proof in https://math.uchicago.edu/~may/REU2020/REUPapers/Pomerantz.pdf, page 5). We also know that $\lim_{n\to\infty} \left(\sum_{k=0}^nA_n \right)$ converges over the 2-adics if and only if $A_n{\to}0_2$ when ${n\to\infty}$.
Now we have that $L=\lim_{n\to\infty} \left(\sum_{k=0}^n3^{-k}2^{\sum_{k=0}^nv_2(f_k)} \right)=\lim_{n\to\infty} \left(\sum_{k=0}^n(3^{-k}\prod_{k=0}^n2^{v_2(f_k)}) \right)$ , where $x$ is any odd 2-adic positive integer, $f_{k+1}(x)=(3f_k / 2^{v_2(f_k)}) + 1$ , $f_0 = 3x +1$ , $v_2(f_k)$ would be the 2-adic valuation of $f_k$, and $L$ must be a positive integer, by condition.
As $x$ is an odd positive integer, we notice that $v_2(f_k) \geq 1$
Given all the conditions above, would it be accurate to say that $L$ converges over the ring of the 2-adic integers? If it does, would it be possible to calculate the value of $L$, even if it was dependant of $x$ (which I guess it might be)?
Thank you in advance for reading. Any comment will be much appreciated.