The Cantor Julia set conjecture for dissipative hyperbolic automorphisms

Let $f$colon ${\mathbb C}^2\to{\mathbb C}^2$ be a dissipative and hyperbolic automorphism, meaning that $|\operatorname{Jac} f|<1$, and suppose that $f$ has no attracting points. Let $J$ denote its Julia set. Cantor Julia set conjecture. Then $J$ is a Cantor set.

This conjecture concerns the disconnected case for hyperbolic polynomial automorphisms of ${\mathbb C}^2$ and seeks to extend the one-dimensional picture in which the absence of attracting behavior leads to a totally disconnected Julia set. Its status is not resolved in the supplied source.