Finite-orbit conjecture for sufficiently large sets of quadratic polynomials

Let $S=\{x^2+c_1,x^2+c_2,\dots,x^2+c_s\}$ be a set of distinct quadratic polynomials with $c_i\in\mathbb{Q}$. A point $P\in\mathbb{Q}$ has finite orbit for $S$ if its forward orbit under all compositions of maps in $S$ is finite. Finite-orbit conjecture for large sets. If $\#S\geq4$, then there are no points $P\in\mathbb{Q}$ with finite orbit for $S$. This is obtained in the paper by combining the preceding conjecture with the stated classification theorem; it extends the unconditional result under the hypothesis that the maximum exact rational period is at most three.