The rational points conjecture for the 3x+1 set

Let $M$ be the map used to embed $\mathbb Z_2$ into $[0,2]$, and let $Q$ be the 3x+1 conjugacy map. Define the 3x+1 set by

$$\mathbf Q_{3x+1}=\{(M(r),M(Q(r))):r\in\mathbb Z_2\}\subset\mathbb R^2.$$

A point of this set has rational coordinates when both coordinates are rational. Rational Points Conjecture. All points in $\mathbf Q_{3x+1}$ have coordinates that are either both rational or both irrational. The source presents this as an immediate reformulation of the Periodicity Conjecture because rationality is preserved by $M$.