This a long comment  might be helpful 

I think the Answer is negative as I pointed out in comment 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 proved 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)