Restrictedness conjecture for Hecke-lisse sheaves with level structure

Let $X$ be a smooth complete curve over $k$, let $\ul{x}$ be a finite set of $k$-points, and let $\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}}$ be the moduli stack of $G$-bundles with full level structure at $\ul{x}$. Let $\operatorname{Shv}_{\operatorname{HL}}(\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}})$ be the category of objects whose Hecke transforms are lisse away from $\ul{x}$. Restrictedness conjecture. The object

$$\ul{\operatorname{Shv}}_{\operatorname{HL}}(\operatorname{Bun}_G^{\operatorname{level}_{\ul{x}}})\in \fL(G)_{\ul{x}}\operatorname{Cat}$$

belongs to $\fL(G)_{\ul{x}}\operatorname{Cat}_{\operatorname{restr}}$. This is a key compatibility needed for the ramified geometric Langlands picture; the source gives no resolution evidence.