Explicit preperiodic-point bounds for quadratic rational maps

Definitions and mathematical statement

Let $f:\mathbb P^1\to\mathbb P^1$ be a degree-two rational map defined over $\mathbb Q$. It has a presentation $f([X:Y])=[F(X,Y):G(X,Y)]$ by homogeneous quadratic forms over $\mathbb Q$ with no common projective zero. The rational projective line includes infinity.

For $P\in\mathbb P^1(\mathbb Q)$ put $\mathcal O_f(P)=\{f^n(P):n\geq0\}$. A point is preperiodic if this orbit is finite, and periodic if $f^n(P)=P$ for some positive $n$; the least such $n$ is its exact period. Let $\operatorname{PrePer}(f,\mathbb Q)$ denote the set of rational preperiodic points.

Do all three bounds hold for every $f$?

1. Every rational periodic point has exact period at most $7$.
2. Every rational preperiodic point has at most $9$ distinct points in its forward orbit, including itself.
3. $\#\operatorname{PrePer}(f,\mathbb Q)\leq14$.

Short English statement

For quadratic rational maps over the rationals, are the universal bounds $7$ on periods, $9$ on finite orbit sizes, and $14$ on all preperiodic points valid?

Sources

R. Benedetto, P. Ingram, R. Jones, M. Manes, J. H. Silverman and T. J. Tucker, Current Trends and Open Problems in Arithmetic Dynamics, arXiv:1806.04980v2 (2018), Conjecture 4.7, https://arxiv.org/html/1806.04980v2. The survey attributes the conjecture to Benedetto, Chen, Hyde, Kovacheva and White (2014), and Blanc, Canci and Elkies (2015). This is an independently written formulation of the three assertions.