Lyubashenko's decomposition conjecture for the automorphic sheaf

Let $X$ be a smooth projective curve and let $E$ be an irreducible $\operatorname{SL}_2$-local system on $X$. Assume Lyubashenko's Conjecture 3 holds for the point ${\cal E}={\cal O}$ of $\operatorname{RCov}^0$; equivalently, suppose there is a $\mathbb Z/2\mathbb Z$-graded complex $SQ_E\in\operatorname{D}(\operatorname{Spec} k)$ with a $\mathbb Z/2\mathbb Z$-graded isomorphism

$$(SQ_E)^{\otimes 2}\simeq CL_E(X).$$

Write $SQ_E^+$ and $SQ_E^-$ for its graded pieces. Lyubashenko's decomposition conjecture. Then

$$F_G(\operatorname{Aut}_E)\simeq SQ_E^+\otimes{\cal F}^+\oplus SQ_E^-\otimes{\cal F}^-,$$

where ${\cal F}^+$ and ${\cal F}^-$ are irreducible perverse sheaves on $\widetilde{\operatorname{Bun}}_G$, and this decomposition is the grading induced by the 2-automorphism $\bar\epsilon$ acting trivially on the auxiliary datum and by $-1$ on the relevant rank-two object. The statement predicts the decomposition of the automorphic sheaf into the two parity pieces determined by the Clifford-algebra square root; its status is unknown in the supplied text.