Byszewski–Ulas digit-sum factorization conjecture

Let $r\ge 1$, let $N_1,\ldots,N_r$ be positive integers, and let $x_1,\ldots,x_r,y_1,\ldots,y_r$ be variables. For each nonnegative integer $n$, write $s_2(n)$ for the sum of the binary digits of $n$. Byszewski–Ulas conjecture. One has

$$\sum_{n_1=0}^{2^{N_1}-1}\cdots\sum_{n_r=0}^{2^{N_r}-1}(-1)^{\sum_{j=1}^r s_2(n_j)}\left(\sum_{j=1}^r s_2(n_j)x_j+n_jy_j\right)^{\sum_{j=1}^r N_j}=(-1)^{\sum_{j=1}^r N_r}\left(\sum_{j=1}^r N_j\right)!\prod_{j=1}^r\prod_{i_j=0}^{N_j-1}(x_j+2^{i_j}y_j).$$

The paper presents this as another conjecture of Byszewski and Ulas and states in the introduction that it proves and generalizes the two displayed conjectures. The identity gives a product factorization for a digit-sum-weighted power sum.