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 and was proven by Allouch which states that ${\mathrm{Col}_{\min}(N) < N}$ For almost all $N$ (in the sense of natural density), and there is an improvement here by Terry Tao in the sense of logarithmic density (see Theorem 2).