Ghys's holomorphic rigidity conjecture for holomorphic Anosov diffeomorphisms

Let $M$ be a compact complex manifold and let $f:M\to M$ be a holomorphic Anosov diffeomorphism. A complex infra-nilmanifold is a quotient of a complex nilmanifold by a finite group of holomorphic affine automorphisms. Ghys's holomorphic rigidity conjecture. Every holomorphic Anosov diffeomorphism is holomorphically conjugate to an automorphism of a complex infra-nilmanifold. Ghys proved the corresponding rigidity statement for complex surfaces, while the conjecture remains open even in dimension $3$.