Finite-order conjecture for rational dynamics of the prime-representing map

Let

$$\mathcal{T}(x)=\left\lfloor x\right\rfloor\bigl(1+\{x\}\bigr),$$

where $\{x\}=x-\left\lfloor x\right\rfloor$. For a rational number $x\geq 2$, define its order under $\mathcal{T}$ to be the least non-negative integer $n$ such that $\mathcal{T}^n(x)$ is an integer, if such an $n$ exists. Finite-order conjecture. Every rational number $x\in\mathbb{Q}_{\geq 2}$ has finite order under $\mathcal{T}$. This conjecture asks whether the rational initial values in the prime-representing recursion always eventually reach an integer; the paper proves density-one results for finite-order fractions at each fixed denominator and completely characterizes the case of denominator $2$, but does not resolve the conjecture in general.