Poonen's conjecture on rational periodic points of quadratic polynomials

Let $N\geq 4$. A quadratic polynomial is a polynomial $f(x)\in\mathbb{Q}[x]$ of degree $2$, and a rational point of period $N$ is an $x\in\mathbb{Q}$ such that $f^{(N)}(x)=x$ but $f^{(m)}(x)\ne x$ for every $1\leq m<N$. Poonen's conjecture. If $N\geq 4$, then there is no quadratic polynomial $f(x)\in\mathbb{Q}[x]$ with a rational point of period $N$. This extends the known non-existence results for rational periodic points of periods $4$ and $5$; the conjecture remains open for all $N\geq 6$.