Chen–Wen–Wu conjecture on generalized Thue–Morse k-binomial complexity

Let $m\geqslant 2$ and let $\mathbf{t}_m$ be the generalized Thue--Morse word over an $m$-letter alphabet. For $k\geqslant 1$, write $\mathsf{b}_{\mathbf{t}_m}^{(k)}(n)$ for the number of $k$-binomial equivalence classes represented by factors of length $n$, where two finite words are $k$-binomially equivalent when every subword of length at most $k$ occurs the same number of times in both words.

Chen–Wen–Wu conjecture. For every $k\geqslant 3$, the function $\mathsf{b}_{\mathbf{t}_m}^{(k)}$ is ultimately periodic with period $m^k$.

The $k$-binomial complexity is known explicitly for the two-letter Thue--Morse word and, for $k=2$, for generalized Thue--Morse words of arbitrary alphabet size. The conjecture proposes the corresponding eventual periodicity for all higher orders; the supplied source does not establish it.