The Flynn–Poonen conjecture on rational cycles of quadratic polynomials

Let $N\geq 4$ and let $f(x)\in{\mathbb Q}[x]$ be a quadratic polynomial. A rational point has exact period $N$ if its least positive period under iteration by $f$ is $N$.

Flynn–Poonen conjecture. There is no quadratic polynomial $f(x)\in{\mathbb Q}[x]$ with a rational point of exact period $N$.

This is a specific uniformity question for rational periodic points of quadratic polynomials, attributed in the source to Flynn and Poonen. The supplied text does not state a resolution, so the conjecture is recorded as open.