Bedford's basin conjecture for contracting automorphisms

Let $\mathbb B_n$ be the unit ball in $\mathbb C^n$, and let $f_1,f_2,\ldots$ be automorphisms of $\mathbb C^n$ fixing the origin. Suppose there are real numbers $a,b$ with $0<a<b<1$ such that, for every $z\in\mathbb B_n$ and $k\in\mathbb N$,

$$a\|z\|\leq\|f_k(z)\|\leq b\|z\|.$$

Bedford's conjecture. The basin of attraction

$$\{z\in\mathbb C^n: F_m(z):=(f_m\circ\cdots\circ f_1)(z)\to0\ \text{as }m\to\infty\}$$

is biholomorphic to $\mathbb C^n$. This is a central basin problem in several complex variables; the source presents it as open.