Generalized Poonen conjecture for unicritical polynomials

For an integer $d\geq 2$ and $c\in\mathbb{Q}$, let $f_{d,c}(z)=z^d+c$. Generalized Poonen conjecture. For $n>3$, there is no $f_{d,c}$ defined over $\mathbb{Q}$ with a $\mathbb{Q}$-rational periodic point of minimal period $n$, and

$$\#\operatorname{Preper}(f_{d,c},\mathbb{Q})\leq 9.$$

The conjecture extends the quadratic case while allowing the degree to vary. The source notes that Narkiewicz resolves the odd-degree special case, whereas the even-degree case remains open.