Becker's conjecture on the Mahler classification of automatic real numbers

Let $\xi$ be an irrational real number whose base-$b$ expansion is generated by a finite automaton; equivalently, its digit sequence is $k$-automatic for some integer base $k\geq 2$. In Mahler's classification, an $S$-number is a real number satisfying $0<w(\xi)<+\infty$, where

$$w(\xi)=\limsup_{n\to\infty}\frac{w_n(\xi)}{n}$$

and $w_n(\xi)$ is the supremum of the real numbers $\omega$ for which infinitely many integer polynomials $P(X)$ of degree at most $n$ satisfy $0<|P(\xi)|\leq H(P)^{-\omega}$. Becker's conjecture. Every irrational automatic real number is an $S$-number. The conjecture asserts that irrational automatic real numbers share the generic Mahler classification of real numbers, since the set of $S$-numbers has full Lebesgue measure. The supplied text does not state whether the conjecture is open or resolved.