This long comment  might be helpful:

I think the answer is negative as I pointed out in the comments because almost all Collatz orbits attain almost bounded values, the result which is shown by [Terras](https://mathscinet.ams.org/mathscinet-getitem?mr=568274) and was proven by [Allouch](https://mathscinet.ams.org/mathscinet-getitem?mr=567874) which states that ${\mathrm{Col}_{\min}(N) < N}$ For almost all $N$ (in the sense of natural density), and there is an improvement [here](https://terrytao.wordpress.com/2019/09/10/almost-all-collatz-orbits-attain-almost-bounded-values/) by Terry Tao in the sense of logarithmic density (**see Theorem 2**).