Bedford's stable-manifold conjecture

Let $X$ be a complex manifold endowed with a Riemannian metric, and let $f\colon X\to X$ be an automorphism acting hyperbolically on an invariant compact subset $K\subset X$. For $p\in K$, let $\Sigma(p)$ denote the stable manifold of $p$, and let $k$ be the stable dimension. Bedford's conjecture. The stable manifold $\Sigma(p)$ is biholomorphic to $\mathbb C^k$.

This conjecture concerns the global complex geometry of stable manifolds arising from hyperbolic dynamics. The source presents it as a well-known conjecture and states that it remains open.