Classification conjecture for three-dimensional holomorphic partially hyperbolic diffeomorphisms

Let $f$ be a holomorphic partially hyperbolic diffeomorphism on a complex $3$-manifold. Three-dimensional classification conjecture. Up to passing to a finite cover, $f$ is holomorphically conjugate to one of the following: an automorphism of a complex torus; a holomorphic skew product over a linear Anosov automorphism; or a time-$t$ map of a holomorphic Anosov flow. The conjecture is motivated by the known low-dimensional rigidity results, but remains open: the corresponding holomorphic Anosov classification is unresolved in dimension $3$, and the skew-product class is also difficult to classify.