No long rational cycles for quadratic maps with a rational critical 2-cycle

Let $\phi$ be a quadratic map defined over $\mathbb{Q}$ with a $\mathbb{Q}$-rational periodic critical point of period $2$. A $\mathbb{Q}$-periodic point of period $n$ is a point in $\mathbb{P}^1(\mathbb{Q})$ whose minimal period under $\phi$ is $n$.

No-long-cycles conjecture. The map $\phi$ has no $\mathbb{Q}$-periodic point of period greater than or equal to $3$.

This conjecture is the hypothesis used in the paper to classify the possible rational preperiodicity graphs for these quadratic maps. The supplied text does not report a proof or disproof.