Lagarias' Periodicity Conjecture for the 3x+1 conjugacy map

Let $\mathbb Z_2$ denote the ring of 2-adic integers, and let $Q:\mathbb Z_2\to\mathbb Z_2$ be the 3x+1 conjugacy map. A 2-adic integer is rational when it belongs to $\mathbb Q$. Lagarias' Periodicity Conjecture. For every $x\in\mathbb Z_2$, $Q(x)$ is rational if and only if $x$ is rational. This conjecture would imply that every rational point of $\mathbb Z_2$ is preperiodic under the 3x+1 map; the supplied text gives no resolution evidence beyond its attribution to Lagarias.