Hutz's conjecture on rational periods of even-degree unicritical polynomials

Let $n>2$ be an integer, let $d>2$ be an even integer, and let $c\in\mathbb Q$. Consider the polynomial map $\varphi_{d,c}$.

Hutz's conjecture. There is no even degree $d>2$ and no $c\in\mathbb Q$ such that $\varphi_{d,c}$ has a rational point of exact period $n$. Moreover,

$$\#\operatorname{PrePer}(\varphi_{d,c},\mathbb Q)\leq 4.$$

The conjecture combines a restriction on rational periodic orbits with a uniform bound on rational preperiodic points. The supplied status evidence records related conditional bounds for sufficiently large even degree, but does not establish this full conjecture unconditionally.