Poonen's rational-period conjecture for quadratic polynomials

Let $\phi(x)=x^2+c$ for some $c\in\mathbb{Q}$, and let $a\in\mathbb{Q}$ be a periodic point for $\phi$, meaning that $\phi^n(a)=a$ for some $n\geq1$; the least such $n$ is the period length of $a$. Poonen's conjecture. If $a$ is a rational periodic point for $\phi$, then its period length is at most $3$. This is a famous conjecture in arithmetic dynamics with partial results, and the paper uses it as an assumption in its rational-coefficient results.