Constant term identity of type $D_2$, I

Let $p$ be the parameter of the $D_2$ identity, let $t$ be an indeterminate, and let $\operatorname{Res}_{x_0,x_1,x_2,x_3,x_4}$ denote the iterated residue in the displayed variables. Constant term identity of type $D_2$, I.

$$\operatorname{Res}_{x_{0},x_{1},x_{2},x_{3},x_{4}}\frac{(1+x_{0})^{6p-2-2t}} {x_{0}^{-8p+3}(x_{1}x_{2}x_{3} x_{4})^{4p}}(1+ x_{1})^{t}(1+ x_{2})^{t}(1+ x_{3})^{t}(1+ x_{4})^{t}(x_{0}-x_{1})^{-4p}(x_{0}-x_{2})^{-4p}(x_{3}-x_{0})^{-4p}\prod\limits_{1\leq i<j\leq 4}(x_{i}-x_{j})^{2p}\frac{\partial_{x_{0}}^{4p-1}}{(4p-1)!} x_{4}^{-1} \delta\left(\frac{x_{0}}{x_{4}}\right)$$ $$=A_p {t + p + 1/2 \choose 4p} {t + 2p \choose 4p -1} {t \choose 4p -1},$$

where $A_p$ is a nonzero constant. The conjecture is cited from earlier work and is introduced because the $m=2$ case of the preceding $D_m$ identity fails; the source gives no general proof.