Flynn–Poonen–Schaefer conjecture on rational periods of quadratic polynomials

Let $n\geq 4$, and let $\varphi_{2,c}(z)=z^2+c$ be a quadratic polynomial with $c\in\mathbb Q$. A rational point has exact period $n$ if its orbit under iteration of $\varphi_{2,c}$ returns to itself for the first time after exactly $n$ iterations.

Flynn–Poonen–Schaefer conjecture. There is no quadratic polynomial $\varphi_{2,c}(z)=z^2+c\in\mathbb Q[z]$ with a rational point of exact period $n$.

This is a special case of the uniform boundedness conjecture for $D=N=1$ and $d=2$. The claim concerns the possible exact periods of rational points and remains open as stated.