2 problems

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

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