A generalized Prouhet–Thue–Morse divisibility conjecture

Let $m\geq 1$ and let $a_{1},\ldots,a_{k}$ be nonnegative integers. Put $A=(a_{1},\ldots,a_{k})$, $X=(x_{1},\ldots,x_{m})$, and $Y=(y_{1},\ldots,y_{m})$. Define

$$G_{A}(X,Y)=\sum_{i_{1}=0}^{k^{a_{1}}-1}\cdots\sum_{i_{k}=0}^{k^{a_{m}}-1}\zeta_{k}^{\sum_{j=1}^{m}s_{k}(i_{j})}\Bigl(\sum_{j=1}^{m}(s_{k}(i_{j})x_{j}+i_{j}y_{j})\Bigr)^{\sum_{j=1}^{m}a_{j}}.$$

Generalized Prouhet–Thue–Morse divisibility conjecture. If $k=2$, then

$$G_{A}(X,Y)=(-1)^{\sum_{j=1}^{m}a_{j}}\Bigl(\sum_{j=1}^{m}a_{j}\Bigr)!\prod_{j=1}^{m}\prod_{i_{j}=0}^{a_{j}-1}(x_{j}+2^{i_{j}}y_{j}).$$

If $k>2$, then

$$\prod_{j=1}^{m}\prod_{i_{j}=0}^{n_{j}-1}(x_{j}+k^{i_{j}}y_{j})\mid G_{A}(X,Y).$$

The conjecture is presented as a further generalization of results in the paper; the supplied text gives no resolution or supporting status evidence.