Friedberg–Ginzburg conjecture on descents and theta representations

For each group and covering degree treated in the paper, let $\sigma_G^{(m)}$ denote the representation generated by the corresponding descent, and let $\Theta_G^{(m)}$ denote the theta representation on the relevant metaplectic cover. Friedberg–Ginzburg conjecture. For each of the descents treated here, the representations coincide:

$$\sigma_G^{(m)}=\Theta_G^{(m)}.$$

The authors have shown that the descents project nontrivially to the corresponding theta representations, but automorphy and irreducibility are not known in general because of an archimedean Whittaker-function issue. The conjecture asserts that the descent representations are exactly the theta representations.