Uniform boundedness conjecture for finite orbits of rational maps

Let $K$ be a number field, let $d$ be an integer, and let $S=\{\phi_1,\phi_2,\dots,\phi_s\}$ be a set of rational maps defined over $K$ such that $2\leq\deg(\phi)\leq d$ for all $\phi\in S$. A point $P\in\mathbb{P}^1(K)$ has finite orbit for $S$ if its forward orbit under all compositions of maps in $S$ is finite. Uniform boundedness conjecture for finite orbits. There exist constants $B=B(K,d)$ and $C=C(K,d)$, depending only on $K$ and $d$, such that if $\#S\geq C$, then there are at most $B$ points $P\in\mathbb{P}^1(K)$ of finite orbit for $S$. Equivalently, for any positive probability measure $\nu$ on $S$, the expected height satisfies

$$\#\big\{P\in\mathbb{P}^1(K):\mathbb{E}_\nu[\hat{h}](P)=0\big\}\leq B.$$

This is proposed as an analogue for sets of maps of the Morton–Silverman conjecture for a single rational map. The conjecture asks for uniform bounds depending only on the number field and the degree bound, while the existence and properties of the relevant expected height are supplied by the surrounding theory.