Möller's trajectory conjecture for the 3x + 1 problem

For any odd positive integer $x_0$, define its trajectory $(x_n)_{n\geqslant 0}$ by choosing positive integers $a_n$ such that

$$x_0=x_0,\qquad x_n=\frac{3x_{n-1}+1}{2^{a_n}}$$

with every $x_n$ odd. A trajectory is pure periodic if it is periodic starting at $x_0$. Möller's trajectory conjecture. (i) $(x_n)_{n\geqslant 0}$ is periodic for every odd positive integer $x_0$; (ii) $(1,1,\ldots)$ is the unique pure periodic trajectory. This is a proposed route to the $3x+1$ problem: periodicity of every trajectory together with uniqueness of the all-ones cycle would rule out nontrivial cycles. The source attributes the approach to Möller; no resolution is supplied here.