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 and was proved 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)