Conjecture on the global integral involving metaplectic theta functions

Let $U_{2r}$ be the unipotent radical of the parabolic subgroup of $GL_{2r}$ with Levi part $GL_2^r$, let $\psi_{U_{2r}}$ be a character whose stabilizer in $GL_2^r$ is the diagonally embedded $GL_2$, and let $E_{\Theta_{2r-1}^{(n)}}$ be the associated Eisenstein series on the $n$-fold cover of $GL_{2r}$. Let $\pi$ be a cuspidal representation of $GL_2^{(k)}(\mathbb{A})$ with trivial central character, and define

$$I(\varphi_\pi,s,f_s)=\int\limits_{Z^n(\mathbb{A})\\,GL_2(F)\backslash GL_2(\mathbb{A})}\int\limits_{U_{2r}(F)\backslash U_{2r}(\mathbb{A})}\varphi_\pi(\mathbf{s}(g))E_{\Theta_{2r-1}^{(n)}}(\mathbf{s}(ug),s,f_s)\psi_{U_{2r}}(u)\\,du.$$

The global-integral conjecture. The integral $I(\varphi_\pi,s,f_s)$ has a meromorphic continuation to the full complex plane, and: (1) if $n<r+1$, it is identically zero for all $s$; (2) if $n=r+1$, it is Eulerian and represents the partial degree-two $L$-function $L^S(\tau(\pi),s)$, where $\tau(\pi)$ is the lift of $\pi$ to $GL_2(\mathbb{A})$; (3) if $n>r+1$, it is not Eulerian and represents a certain Dirichlet series which, in the domain $\operatorname{Re}(s)>1/2$, can have at most a simple pole at $s=\frac{n+1}{2n}$. This conjecture organizes the expected behavior of the integral according to the relation between the cover degree $n$ and the parameter $r$; the paper proves the relevant Eulerian calculation in the case $r=2$, $n=3$, while the general assertions remain open in the source.