Uniform boundedness conjecture for common preperiodic points of rational maps

Let $d\geq 2$, and let $f,g\in\mathbb{C}(z)$ be rational functions of degree $d$. Write $\operatorname{Preper}(f)$ and $\operatorname{Preper}(g)$ for their sets of preperiodic points.

Uniform boundedness conjecture. For each degree $d\geq 2$, there exists a constant $B=B(d)$ such that either

$$|\operatorname{Preper}(f)\cap\operatorname{Preper}(g)|\leq B$$

or

$$\operatorname{Preper}(f)=\operatorname{Preper}(g)$$

for every pair of rational functions $f$ and $g$ in $\mathbb{C}(z)$ of degree $d$.

For rational maps of degree at least two, the intersection of the preperiodic sets is already known to be either finite or equal to the full preperiodic set; the conjecture asks for a bound depending only on the common degree in the finite case. The paper proves an analogous uniform bound for distinct quadratic polynomials, but the stated conjecture for arbitrary rational maps remains open.