Rationality criterion for admissible 2-adic series

Let $\overline{\bf N}$ denote the codomain of an admissible function $\psi:{\bf N}\to\overline{\bf N}$, and define the power series

$$G_\psi(T)=\sum_{n=0}^{\infty}2^{\psi(n)}T^n.$$

Rationality criterion. The power series $G_\psi(T)$ is rational if and only if $G_\psi(\frac{1}{3})$ is a rational element of ${\bf Q}_2$. This is presented as a reformulation of the Collatz conjecture in the paper. The supplied text does not provide an independent resolution of this criterion, so it remains open.