The regular-word existence and uniqueness conjecture for generalized Thue–Morse sequences

Let $r\geqslant 2$ and let $n$ be the length of a word over a two-letter alphabet. An $r$-regular word is a word satisfying the corresponding regularity condition, and $\operatorname{PTE}(2^{r+1},2,r)$ denotes the set of words representing Prouhet–Tarry–Escott solutions with these parameters. Regular-word existence and uniqueness conjecture. On a two-letter alphabet, there are $r$-regular words of length $n$ if and only if

$$n=k\cdot 2^r,\qquad k\geqslant 2.$$

Moreover, $\operatorname{PTE}(2^{r+1},2,r)$ contains just one word: the initial segment of the Thue–Morse sequence of length $2^{r+1}$. The preceding theorem establishes existence whenever $n=k\cdot 2^r$ with $k\geqslant 2$; the conjectural content is the converse for all $r\geqslant 2$ and the asserted uniqueness of the corresponding Prouhet–Tarry–Escott word.