Byszewski–Ulas digit-sum summation conjecture for powers of binary sums

Let $r\ge 1$, let $N_1,\ldots,N_r$ be positive integers, and let $x,y_1,\ldots,y_r$ 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

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

The paper states that this conjecture is proved and generalized there; the displayed identity concerns weighted power sums arising from the binary digit-sum function.