The geometric Langlands conjecture for irreducible local systems

Let $k$ be a field, let $X$ be a smooth projective curve over $k$, and let $G$ be a reductive group over $k(X)$ with Langlands dual group $\LG$. Let $S\subset X$ be a finite set. For every irreducible $\LG$-local system $E$ on $X-S$, there exists a non-zero perverse sheaf $\sA=\sA_E$ on the moduli of $G$-bundles on $X$, equipped with appropriate level structure on $S$. The geometric Langlands conjecture. The perverse sheaf $\sA$ is a Hecke eigensheaf with eigenvalue $E$. This is a core conjecture of the geometric Langlands program, which seeks a duality between moduli of $\LG$-local systems on $X$ and moduli of $G$-bundles on $X$. The paper's abstract says that the authors construct the relevant Hecke eigensheaves for irreducible hypergeometric local systems, but the general statement above is not established here.