Constant term identity of type $A_m$, I

Let $m$ and $p$ be the parameters of the $A_m$ constant-term identity, let $t$ be an indeterminate, and let $\operatorname{Res}_{x_0,x_1,\ldots,x_{m+1}}$ denote the iterated residue in the displayed variables. Constant term identity of type $A_m$, I.

$$\operatorname{Res}_{x_0,x_1,\dots,x_{m+1}}\frac{(1+x_0)^{2p-1-t} \prod\limits_{i=1}^{m+1} (1+x_i)^t}{x_0^{2+2p} (x_1 \cdots x_{m+1})^{2mp}} \prod\limits_{i=1}^{m+1} \left(1-\frac{x_i}{x_0}\right)^{-2p} \prod\limits_{1 \leq i < j \leq m+1}(x_i-x_j)^{2p}$$ $$=\lambda_{p,m} {t+ mp \choose 2(m+1)p-1} \prod\limits_{i=1}^{m-1} {t+(i-1)p \choose 2ip-1},$$

where $\lambda_{p,m}\neq0$. This is one of the constant-term identities on which the paper's orbifold-module classification program is based; the source reports it as a conjecture and gives no resolution.