The odd cycles conjecture for the 3x+1 conjugacy map

Let $Q$ be the 3x+1 conjugacy map on $\mathbb Z_2$, and call a periodic orbit an odd cycle when all its elements are odd. Odd Cycles Conjecture. The function $Q$ has exactly two odd fixed points, $-1$ and $1/3$, and exactly two odd cycles of prime period $2$, namely

$$\left(-\frac{1}{3},1\right)\quad\text{and}\quad\left(-\frac{1}{5},\frac{5}{7}\right).$$

There exists no other odd cycle, rational or not. The claim is motivated by the paper's numerical search, which found no $Q$-cycle of prime period greater than $2$.