Geometric Langlands conjecture for generic normalized eigensheaves

Let $X$ be the curve and $G$ the reductive group of the paper. Let $E$ be a $\check{G}$-local system on $X$, and put $W=V^{\gamma}_E$. Assume that $W$ is irreducible and that whenever a $\check{G}$-local system $E'$ satisfies $V^{\gamma}_{E'}\simeq V^{\gamma}_E$, one has $E'\simeq E$. Let $d_G$ be the dimension shift used in the paper, let $\pi:{\cal Y}_d\to\operatorname{Bun}^d_G$ be the projection, let $\mathcal P^d_{W,\psi}$ be the indicated perverse-sheaf complex, and let $\mathbb D K$ denote Verdier duality. Geometric Langlands conjecture. There exists $N>0$ and, for every $d\geq N$, a nonempty open substack $U_d\subset\operatorname{Bun}^d_G$ and an $E$-Hecke eigensheaf $K$ on $\operatorname{Bun}_G$ such that both $K$ and $\mathbb D K$ are generic normalized; $\pi_!\mathcal P^d_{W,\psi}|_{U_d}$ is in perverse degrees at most $d-d_G$, the map from the source induces an isomorphism $\mathcal H^{d-d_G}(\pi_!\mathcal P^d_{W,\psi})\simeq K|_{U_d}$ on the top perverse cohomology sheaves; and $K$ is irreducible on every $\operatorname{Bun}^d_G$ and does not vanish over $U_d$. This predicts the existence and generic behavior of the desired geometric Langlands eigensheaf under the stated irreducibility and rigidity condition. The source gives no resolution, so the conjecture remains open.