Lagarias's periodicity conjecture for the 2-adic Collatz map

Let $\mathbb{Z}_2$ be the ring of $2$-adic integers, let $\overline{T}\colon\mathbb{Q}_2\to\mathbb{Q}_2$ be given by

$$\overline{T}_v(f)=\frac{3^v f+v}{2}\qquad(v\in\{0,1\}),$$

where at each iterate $v$ is the residue modulo $2$ of the current element. An element of $\mathbb{Z}_2$ is eventually $\overline{T}$-periodic if some iterate of it is periodic under $\overline{T}$. Lagarias's periodicity conjecture. The eventually $\overline{T}$-periodic elements of $\mathbb{Z}_2$ are exactly the elements of $\mathbb{Q}\cap\mathbb{Z}_2$. It is known that all eventually $\overline{T}$-periodic elements are rational, while the converse is the conjecture attributed to Lagarias. The conjecture remains open.