Kadell's first constant-term conjecture for the Dyson product

Let $I=\{i_1,\ldots,i_m\}$ be a proper subset of $\{0,1,\ldots,n\}$ and let $J=\{j_1,\ldots,j_m\}$ be a multi-subset of $\{0,1,\ldots,n\}\setminus I$, where $0\leqslant i_1<\cdots<i_m\leqslant n$ and $0\leqslant j_1\leqslant\cdots\leqslant j_m\leqslant n$. Let $a=a_0+a_1+\cdots+a_n$, and let $\operatorname{CT}_{\mathbf{x}}$ denote the constant-term operator in $\mathbf{x}=(x_0,\ldots,x_n)$. Kadell's first conjecture. For nonnegative integers $a_0,a_1,\ldots,a_n$,

$$\left(1+a-\sum_{k\in I}a_k\right)\operatorname{CT}_{\mathbf{x}}\prod_{k=1}^m\left(1-\frac{x_{j_k}}{x_{i_k}}\right)\prod_{0\leqslant i\ne j\leqslant n}\left(1-\frac{x_i}{x_j}\right)^{a_i}=\left(1+a\right)\frac{a!}{a_0!a_1!\cdots a_n!}.$$

This is the first of two conjectures of Kadell concerning constant terms of Dyson products; the paper's stated objective is to prove it, so it is resolved in the source, while the broader constant-term identities relate to the Dyson conjecture and its $q$-analogue.