Is every irrational number containing only $2$ distinct digits, transcendental?

Question

If we have an irrational number, consisting of only $2$ distinct digits, for example:

$$0.01011011101111011111 \cdots$$

Can we conclude that the number is transcendental?

It is conjectured that every irrational algebraic number is normal in base $10$. This would imply that the answer to my question is yes. But can we prove it?

Answer 1

You may want to look at this answer in mathoverflow. Our conditions are not strong enough to use the theorem though, as we just have $c_x(n)\leq 2^n$.