Conjecture on the unipotent Fourier coefficient of the higher theta representation

Let $F$ be the global field and let $\theta_{2r}^{(r+1)}$ denote the higher theta representation on the $(r+1)$-fold cover of $GL_{2r}(\mathbb{A})$. For $g$ in the embedded $GL_2^{(r+1)}(\mathbb{A})$, define

$$f(g)=\int\limits_{U_{2r}(F)\backslash U_{2r}(\mathbb{A})}\theta_{2r}^{(r+1)}(ug)\psi_{U_{2r}}(u)\,du.$$

Fourier-coefficient conjecture. The automorphic representation of $GL_2^{(r+1)}(\mathbb{A})$ generated by all the functions $f(g)$ is $\Theta_2^{(r+1)}$. The conjecture predicts the precise automorphic representation generated by this Fourier coefficient; the source presents it as a possible application of the preceding global-integral conjecture, and no resolution is supplied here.