Kadell's orthogonality conjecture

Let $a=(a_0,\dots,a_n)$ be a sequence of non-negative integers, let $x^{(a)}$ be the alphabet defined by

$$x^{(a)}=(x_0,x_0q,\dots,x_0q^{a_0-1},\dots,x_n,x_nq,\dots,x_nq^{a_n-1}),$$

and write $|a|=a_0+\cdots+a_n$. For a composition $v=(v_0,\dots,v_n)$, write $|v|=v_0+\cdots+v_n$, and let $D_{v,(r)}(a)$ be the generalized $q$-Dyson constant term defined in the source. Kadell's conjecture.** For $r$ a positive integer and $v$ a composition such that $|v|=r$,

$$D_{v,(r)}(a)=\begin{cases} \displaystyle \frac{q^{\sum_{i=k+1}^n a_i}(1-q^{a_k})(q^{|a|};q)_r}{(1-q^{|a|})(q^{|a|-a_k+1};q)_r}\prod_{i=0}^n\genfrac{[}{]}{0pt}{}{a_i+\cdots+a_n}{a_i},&\text{if }v=(0^{k},r,0^{n-k}),\\[6mm] 0,&\text{otherwise}. \end{cases}$$

Kadell formulated this orthogonality conjecture as a $q$-Dyson constant-term problem. The more general statement in the source was proved by Károlyi, Lascoux and Warnaar using multivariable Lagrange interpolation and key polynomials, so this conjecture is solved.