The conjecture that the common preperiodic-point bound equals $4d-1$

For each $d\geq 2$, let $B_d$ denote a uniform bound such that every pair $f,g\in\operatorname{Rat}_d$ satisfies either

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

or

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

The $4d-1$ bound conjecture. One can take

$$B_d=4d-1.$$

The preceding results show that any general bound must be at least $4d-1$, while examples give larger finite intersections in every degree. Thus the claim identifies the conjecturally optimal uniform bound for pairs of degree-$d$ rational maps.