Genericity conjecture for the automorphic sheaf constructed by theta lifting

In the situation of Theorem~, let $F_G(K_{E,\chi, \mathbb H})$ be the complex on $\operatorname{Bun}_{\mathbb G}$ constructed there, and let $\Whit$ denote the first Whittaker coefficients functor. The complex has the form

$$F_G(K_{E,\chi, \mathbb H})={\mathcal K}\otimes {\mathcal E},$$

where $\mathcal E$ is a constant complex and $\mathcal K$ is a perverse sheaf irreducible on each connected component of $\operatorname{Bun}_{\mathbb G}$. Genericity conjecture. The first Whittaker coefficient identifies with

$$\Whit({\mathcal K})\simeq \mathbb{\bar Q}_\ell$$

up to a cohomological shift. This predicts that the automorphic sheaf arising in Theorem~ is irreducible on each connected component and has a nonzero, one-dimensional first Whittaker coefficient; the supplied text gives no evidence that the assertion has been proved or refuted.