Uniform recurrence conjecture for the Thue–Morse two-block fixed point

Let $x^{(00)}$ be the fixed point of the Thue–Morse two-block substitution. A word is uniformly recurrent when every finite factor occurring in $x^{(00)}$ occurs there with bounded gaps. Uniform recurrence conjecture. $x^{(00)}$ is uniformly recurrent. Uniform recurrence is presented as one of the conjectures related to the difficulty of understanding the language of this fixed point, alongside questions about word frequencies.