Whittaker nonvanishing conjecture for the theta representation of $Sp_{r-1}^{(r)}$

Let $r>1$ be odd, let $F$ be a global field, and let $\Theta_{r-1}^{(r)}$ be the theta representation of the $r$-fold metaplectic cover of $Sp_{r-1}({\mathbb A})$. Put $r_1=(r-1)/2$ and, for $\delta\in F^*$, let

$$e_{\delta}(u)=\psi(u_{1,2}+\dots+u_{r_1-1,r_1}+\delta u_{r_1,r_1+1})$$

be the corresponding Whittaker character. Orbit Conjecture. For each class in $F^*/(F^*)^2$, there is a representative $\delta\in F^*$ and a function $\theta$ in $\Theta_{r-1}^{(r)}$ such that the Whittaker integral of $\theta$ with respect to $e_{\delta}$ is nonzero. The usual genericity assertion is a consequence of the orbit conjecture, and this stronger nonvanishing statement requires nonvanishing in every square class; no resolution is given here.