@stargazer07817 just posted a great paper in Unicity of trajectory - #8 by stargazer07817 (https://math.colgate.edu/~integers/t8/t8.pdf) which considers this sequence going “the other way” as 2-adic integers (Z_2) and this representation has many interesting properties. They extend the Collatz map from the regular integers to 2-adic integers and then define the function Q: Z_2 \to Z_2 which maps a 2-adic integer to it’s parity sequence. Q is a bijection (generalizing the “Unicity of trajectories” result) and even an isometry (in 2-adic metric) which is basically equivalent to the result that the first n bits of the parity sequence are determined precisely by the lowest n bits of the starting number.
IIUC, all eventually periodic 2-adic numbers are “rational” numbers (in the sense that if you multiply them by integers you get integers). So the same observation applies here. From that paper it seems like they reference a conjecture that is broader than the Collatz conjecture that Q maps all rational numbers to rational numbers (Collatz conjecture is that it maps all integers (eventually terminating numbers) to rationals (eventually periodic numbers)).