Ramified global geometric Langlands correspondence

Let $\oX=X-\ul{x}$, let $\operatorname{LS}^{\operatorname{restr}}_{cG}(\oX)$ and $\operatorname{LS}^{\operatorname{restr}}_{cG}(\ocD_{\ul{x}})$ be the restricted moduli stacks of local systems on the punctured curve and formal punctured discs, respectively, and let $\operatorname{fr}$ be restriction from the former to the latter. Let $\operatorname{IndCoh}_\Nilp(\operatorname{LS}^{\operatorname{restr}}_{cG}(\oX)\mid\operatorname{LS}^{\operatorname{restr}}_{cG}(\ocD_{\ul{x}}))$ be the corresponding relative ind-coherent object. Ramified global geometric Langlands conjecture. Under the conjectural local equivalence

$$\fL(G)_{\ul{x}}\operatorname{Cat}_{\operatorname{restr}}\simeq \operatorname{tIndCoh}_\Nilp(\operatorname{LS}^{\operatorname{restr}}_{cG}(\ocD_{\ul{x}})),$$

the object $\ul{\operatorname{Shv}}_{\operatorname{HL}}(\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}})$ corresponds to

$$\operatorname{IndCoh}_\Nilp(\operatorname{LS}^{\operatorname{restr}}_{cG}(\oX)\mid\operatorname{LS}^{\operatorname{restr}}_{cG}(\ocD_{\ul{x}})).$$

This is the spectral description of the Hecke-lisse automorphic object in the ramified setting; it remains conditional on the local equivalence and the restrictedness conjecture.