The Beilinson–Drinfeld regular-singularity conjecture for Hecke eigensheaves

Let $G$ be a complex reductive group, let $X$ be a complex curve, and let $\lambda$ be a ${}^LG$-oper on $X$. Denote by $\Delta^0_\lambda$ the corresponding Hecke eigensheaf or twisted $D$-module on $\operatorname{Bun}_G$.

Beilinson–Drinfeld regular-singularity conjecture. The $D$-module $\Delta^0_\lambda$ is irreducible and has regular singularities on each connected component of $\operatorname{Bun}_G$.

The source identifies this as a conjecture of Beilinson and Drinfeld and uses it to deduce one-dimensionality of suitable eigenspaces. It is presented as an open question in the paper.