Bump-Friedberg generating-function identity for covering groups

Let $(\widetilde{\operatorname{GL}}_r^{(n)},\widetilde{\operatorname{GL}}_R^{(n)})$ be a fundamental pair with $n\mid(\mathbf{p}\cdot n_\alpha)$. Let $\Theta(\widetilde{\operatorname{GL}}_R^{(n)},\chi)$ be the distinguished theta representation whose associated character satisfies $\nu_R=\mathbf{1}$, and let $\mathcal{W}^{\operatorname{GL}_R}_{\mathcal{O}_0}$ be its unique Whittaker model. Let $\widetilde{\Delta}_s$ be the anti-genuine, $K_r$-biinvariant function on $\widetilde{\operatorname{GL}}_r^{(n)}$ specified by the stated torus support and value. Bump-Friedberg identity. For every $\widetilde{g}\in\widetilde{\operatorname{GL}}_r^{(n)}$,

$$\int_{U_r}\widetilde{\Delta}_{n_\alpha s-\frac{R-1}{2}}(u\widetilde{g})\psi(u)\,du=|\det(\widetilde{g})|^{s-\frac{R-r}{2}}\overline{\mathcal{W}^{\operatorname{GL}_R}_{\mathcal{O}_0}}(\phi(\widetilde{g})).$$

This identity is presented in the supplied text as a consequence of the generating-function approach of Bump and Friedberg to the Bump-Hoffstein conjecture; no resolution evidence is supplied.