Generic hyperbolicity conjecture for complex Hénon maps with dominated splitting

A complex Hénon map is a biholomorphic polynomial automorphism of $\mathbb{C}^2$, and it has dominated splitting on its relevant invariant set when the tangent dynamics admits a dominated invariant splitting. A map is Kupka–Smale when its periodic points and invariant manifolds satisfy the usual Kupka–Smale transversality conditions.

Generic hyperbolicity conjecture. Generically, Kupka–Smale complex Hénon maps with dominated splitting are hyperbolic.

This conjecture is motivated by the corresponding result for real maps with dominated splitting, where the dynamically defined center-unstable leaf condition requires a Kupka–Smale hypothesis to imply hyperbolicity. The source does not specify the topology or precise meaning of genericity, so its resolution remains open here.