Bedford's conjecture on basins of attraction of non-autonomous automorphisms

Let $(\Phi_{n,n+1})_{n\in\mathbb N}$ be a family of automorphisms of $\mathbb C^q$. Assume that there exist $0<\nu\leq\mu<1$ and $r>0$ such that

$$\nu|z|\leq |\Phi_{n,n+1}(z)|\leq \mu|z|,\quad z\in r\mathbb B^q,\ n\in\mathbb N.$$

The basin of attraction

$$\{z\in\mathbb C^q: \lim_{n\to\infty}\Phi_{n-1,n}\circ\cdots\circ\Phi_{0,1}(z)=0\}$$

is Bedford's conjecture. biholomorphic to $\mathbb C^q$. The conjecture asks whether every basin arising from such a uniformly contracting sequence of automorphisms is biholomorphic to affine space; the surrounding discussion notes that the corresponding basin is known to be Stein, Runge, Kobayashi-degenerate, and diffeomorphic to $\mathbb C^q$, while biholomorphic equivalence remains open when $\ell\geq2$.