The conjecture has been neither refuted nor proved. The state of the art, as far as I know, is contained in the papers of <a href="http://annals.math.princeton.edu/2007/165-2/p04">Adamczewski and Bugeaud</a>, in which they show that anything with a very low complexity decimal expansion cannot be algebraic. The complexity is the function $c_x(n)$ giving the number of blocks of length $n$ in the decimal expansion of $x$ (or any base). They show that if there exists a $k$ such that $c_x(n)\le kn$ for all $n$, then $x$ is either rational or transcendental. Of course, it's conjectured that $c_x(n)=10^n$ for all algebraic irrationals $x$. Your condition would be implied by the conjecture $c_x(n)>9^n$ for all algebraic irrationals $x$.