Constant term identity of type $D_m$, $m>2$

Let $m>2$, let $p$ be the parameter of the $D_m$ identity, let $t$ be an indeterminate, and let $\operatorname{Res}_{x_0,x_1,\ldots,x_{m+2}}$ denote the iterated residue in the displayed variables. Constant term identity of type $D_m$, $m>2$.

$$\operatorname{Res}_{x_{0},x_{1},\dots, x_{m+2}}\frac{(1+x_{0})^{m^2p+mp-(m+1)t}} {x_{0}^{-4mp+3}(x_{1}\cdots x_{m+2})^{4p}}\prod\limits_{i=1}^{m+2}(1+ x_{i})^{t}(x_{0}-x_{m+1})^{-2mp}(x_{0}-x_{m+2})^{-2mp}\prod\limits_{1\leq i<j\leq m+2}(x_{i}-x_{j})^{2p}\prod\limits_{i=1}^{m}(x_{i}-x_{0})^{-2mp}$$ $$=-\operatorname{Res}_{x_{0},x_{1},\dots, x_{m+2}}\frac{(1+x_{0})^{m^2p+mp-(m+1)t}} {x_{0}^{-4mp+4}(x_{1}\cdots x_{m+2})^{4p}}\prod\limits_{i=1}^{m+2}(1+ x_{i})^{t}(x_{0}-x_{m+1})^{-2mp}(x_{0}-x_{m+2})^{-2mp}\prod\limits_{1\leq i<j\leq m+2}(x_{i}-x_{j})^{2p}\prod\limits_{i=1}^{m}(x_{i}-x_{0})^{-2mp}$$ $$=\alpha_{p,m} {t+p+\tfrac{1}{2} \choose 2p} {t-p+ \tfrac{1}{2} \choose 2p} {t-(m+1)p+1 \choose p} {t+mp \choose p} \prod\limits_{l=0}^{m-1}{t+pl \choose (m+1)p-1},$$

where $\alpha_{p,m}$ is a nonzero constant. The source reports verification for $m=3$, $p\leq4$ and $m=4$, $p\leq2$, while noting that the analogous conjecture does not hold for $m=2$.