Rational period conjecture for quadratic polynomials

Let $N$ be an integer with $N\geq 4$, and let $f(z)\in\mathbb{Q}[z]$ be a quadratic polynomial. A point is of exact period $N$ if its forward orbit under $f$ has least period $N$. Rational period conjecture. If $N\geq 4$, then there is no quadratic polynomial $f(z)\in\mathbb{Q}[z]$ with a rational point of exact period $N$.

This is a basic case of the conjectural classification of rational preperiodic points for quadratic polynomials over $\mathbb{Q}$. The paper studies the resulting classification under this assumption; the source does not provide evidence that the conjecture has been resolved.