Conjecture on the frequency of 1 in the Thue–Morse two-block fixed point

Let $x^{(00)}$ be the Thue–Morse two-block fixed point, and let the frequency of $1$ mean the limiting proportion of positions occupied by $1$ in its prefixes, if that limit exists. Frequency-one-half conjecture. The frequency of $1$ in $x^{(00)}$ exists and equals

$$\frac{1}{2}.$$

This is a special case of the source's broader conjectural programme concerning existence and values of factor frequencies.