On a much weaker version of the Normal conjecture

I would like to ask you about the following question. It is conjectured that every algebraic irrational number is normal (absolutely normal). I know the result by Bugeaud and Adamczewski about the non-linearity of the complexity function. Unfortunately, this was not enough to solve the question I have in mind. But, my question is the following:

Any algebraic irrational number contains at least one 0 in its expansion in basis $$b$$ for all $$b\geq 2$$ sufficiently large (or at least for $$b$$ in the form $$10^s$$, for every $$s$$ sufficiently large)?

• Do you really mean base $b$ rather than $b$-adic? – Robert Israel Jan 8 at 4:48
• @RobertIsrael thanks! – Jeremy Jan 8 at 5:01
• Are you asking whether this assertion has been proved? To my knowledge, it has not. – Greg Martin Jan 8 at 8:44
• Do you know this article by Michel Waldschmidt: arxiv.org/abs/0908.4034? Look at Conjecture 1.1. – Kurisuto Asutora Jan 8 at 8:48
• I'm pretty sure it is not known whether, say, $\sqrt{2}$ contains infinitely many zeros in its decimal expansion. If not, you could very easily make a quadratic irrational with no zeros past the decimal point. – Wojowu Jan 8 at 12:26