The 3x+1 problem in terms of the conjugacy map

Let $\mathbb Z^+$ denote the positive integers, let $\mathbb Z_2$ be the 2-adic integers, and let $Q$ be the 3x+1 conjugacy map. The 3x+1 problem. One should have

$$Q\left(\mathbb Z^{+}\right)\subset\frac{1}{3}\mathbb Z,$$

or equivalently,

$$\mathbb Z^{+}\subset Q^{-1}\left(\frac{1}{3}\mathbb Z\right).$$

This is a reformulation of the original 3x+1 problem through $Q$; the supplied status evidence indicates that the related reverse inclusion in the paper's conjectural discussion fails.