Proportion of a digit in an algebraic number's binary expansion

Question

We know that

1. A real number is rational if and only if it's binary (or base $n$ expansion, for all $n$) is eventually periodic. Therefore, the proportion of each digit (0 or 1 in the binary case) is a rational number.
2. If the asymptotic proportion of each digit is well-defined and rational, we can still easily construct such a number which is not rational.
3. If $x$ is transcendental, the proportion of digits in $x$ might still be algebraic, because almost every number is normal.

But what about

4. If $x$ is algebraic, does that mean that the proportion of each digit in the binary expansion is algebraic?