The degree conjecture for multivariate Prouhet–Thue–Morse polynomials

Let $m,n\geq 1$ be integers, let $t$ be a variable, and put $X=(x_{1},\ldots,x_{m})$. Define

$$H_{m,n}(X,t)=\sum_{i_{1}=0}^{2^{n}-1}\cdots\sum_{i_{m}=0}^{2^{n}-1}(-1)^{s_{2}(\sum_{j=1}^{m}i_{j})}\Bigl(t+\sum_{j=1}^{m}i_{j}x_{j}\Bigr)^{n}.$$

Degree conjecture. One has $\operatorname{deg}_{t}H_{m,n}(X,t)=m-1$. In particular,

$$H_{1,n}=(-1)^{n}n!2^{\frac{n(n-1)}{2}}x_{1}^{n},$$

and

$$H_{2,n}=(-1)^{n}n!2^{\frac{n(n-1)}{2}}\left(2\frac{x_{1}^{n}-x_{2}^{n}}{x_{1}-x_{2}}t+2^{n}\frac{x_{1}^{n+1}-x_{2}^{n+1}}{x_{1}-x_{2}}+x_{1}x_{2}(2^{n}-1)\frac{x_{1}^{n-1}-x_{2}^{n-1}}{x_{1}-x_{2}}\right).$$

The conjecture extends the paper’s results from Section 3 to sums involving several indices. The supplied text does not indicate whether the degree assertion has been proved or disproved.