Uniform periodic-point bound for Hénon-type polynomial automorphisms

Let $f$ be a complex polynomial automorphism of Hénon type such that $|\operatorname{Jac}(f)|\ne 1$. For an algebraic curve $C$, write $\deg(C)$ for its degree, and write $\deg(f)$ and $\operatorname{Jac}(f)$ for the degree and Jacobian of $f$. Uniform boundedness conjecture. The cardinality of the set of periodic points of $f$ lying on $C$ is bounded above by a constant depending only on $\deg(C)$, $\deg(f)$, and $\operatorname{Jac}(f)$. The paper confirms a weaker form of this conjecture and explains that automatic uniformity results provide additional motivation; the stated stronger uniform bound remains open in the supplied text.