Topological conjugacy invariance of hyperbolicity for polynomial automorphisms of the plane

Let $f_0$ and $f_1$ be polynomial automorphisms of ${\mathbb{C}^2}$ with non-trivial dynamics. Write $J_i=J(f_i)$ and $J_i^\star=J^\star(f_i)$ for their Julia sets and assume that $f_0$ is hyperbolic. Suppose there are neighborhoods $N_0$ and $N_1$ of $J_0=J_0^\star$ and $J_1^\star$, respectively, and a homeomorphism

$$\phi:N_0\to N_1$$

such that $\phi\circ f_0=f_1\circ\phi$ wherever these compositions make sense. Topological conjugacy invariance conjecture. Then $f_1$ is hyperbolic.

The conjecture asks whether hyperbolicity is invariant under a local topological conjugacy near the Julia sets, analogous to the corresponding one-dimensional criterion. It is refuted: the claim fails when the conjugating homeomorphism is defined only on $J_0=J_0^\star$.