The Descent Conjecture for the metaplectic theta representation

Let $r$ be odd, and let $\rho$ be the representation of $Sp_{2n+r-2k+1}^{(2r)}({\mathbb A})$ generated by the descent functions defined by the integral construction in the paper. Descent Conjecture. The representation $\rho$ is in the residual spectrum, and each irreducible summand of $\rho$ is the representation $\Theta_{2n+r-2k+1}^{(2r)}$. A preceding proposition gives that, assuming the Orbit Conjecture, $\rho$ is a subrepresentation of the relevant Hilbert space and has nonzero projection to the residual spectrum. The stronger assertion identifying every irreducible summand is not established in the paper.