Periodicity conjecture for the k-binomial complexity of generalized Thue–Morse words

Let $m\geq 3$, and let $\mathbf{t}_{m}$ be the generalized Thue–Morse word. For $k\geq 3$, write $b_{\mathbf{t}_{m},k}(n)$ for its $k$-binomial complexity.

Periodicity conjecture. For every $k\geq 3$, the function $b_{\mathbf{t}_{m},k}(n)$ is ultimately periodic with period $m^{k}$.

The cases $k=1$ and $k=2$ are known to be ultimately periodic with periods $m$ and $m^{2}$, respectively; the conjecture proposes the analogous period $m^{k}$ for every higher binomial complexity.