Uniform Boundedness Conjecture for quadratic polynomials

Let $f_c(x)=x^2+c\in\mathbb{Q}[x]$ be a quadratic polynomial, and let a rational point have exact period $n$ if its period is $n$ and no smaller positive period. Uniform Boundedness Conjecture for $x^2+c$. If $n\geq 4$, then no such polynomial has a rational point of exact period $n$. This is the quadratic-polynomial case of uniform boundedness. The source states that the cases $n=4$ and $n=5$ are proved, while the case $n=6$ is known conditionally on the Birch and Swinnerton-Dyer conjecture for a certain genus-four curve.