The dynamical Manin–Mumford conjecture for complex polynomial automorphisms

Let $f$ be a complex polynomial automorphism of Hénon type of the affine plane, and let $C$ be an irreducible algebraic curve containing infinitely many periodic points of $f$. An involution $[?]$ is an automorphism $[?]$ satisfying $[?]^2=\operatorname{id}$. Dynamical Manin–Mumford conjecture. There exists an involution $\sigma$ of the affine plane whose set of fixed points is $C$ and an integer $n\ge 1$ such that

$$\sigma f^n \sigma=f^{-n}.$$

Reversible polynomial automorphisms provide the motivating examples: intersections of the fixed curve of a reversing involution with its iterated inverse images yield infinitely many periodic points. The conjecture gives a converse classification for curves containing infinitely many periodic points; the supplied text does not state a resolution.