Equality conjecture for the residual and theta representations of metaplectic symplectic groups

Let $r$ be odd with $r<2n$, and let $\sigma_{2n-r+1}^{(2r)}$ and $\Theta_{2n-r+1}^{(2r)}$ denote the representations constructed in the paper on the metaplectic symplectic group of rank $2n-r+1$. Equality conjecture. The representation $\sigma_{2n-r+1}^{(2r)}$ is equal to $\Theta_{2n-r+1}^{(2r)}$. This stronger statement is presented after a proposition showing nonvanishing and residual-spectrum properties of $\sigma_{2n-r+1}^{(2r)}$, and is motivated by analogous results in the cited work; its resolution is not given in the supplied text.