Generalize the coordinate-transformation isomorphism Φ to varying G-bundles over the moduli stack

Determine whether the coordinate-transformation isomorphism Φ, defined for a fixed G-bundle P as an isomorphism of twisted D-modules Φ: p_{((z,v),h_z)}^*Δ^{Q}_{oldsymbol{K}(oldsymbol{M})} → H_λ(Δ^{P}_{oldsymbol{K}(oldsymbol{M})})|_{C^{P}_{oldsymbol{K},((z,v),h_z)}, can be extended to allow variation of the underlying G-bundle across the moduli stack Bun_{G,oldsymbol{K}}^X; specifically, construct a generalization of Φ over a substack of Bun_{G,oldsymbol{K}}^X that uniformly realizes the Hecke-modified (N+1)-point coinvariants via coordinate transformations when P varies.

Background

The paper proves that Hecke modifications of conformal blocks at non-critical level are equivalent to inserting a twisted vacuum module and provides an explicit isomorphism Φ, via coordinate transformations, between the Hecke-modified N-point coinvariants and (N+1)-point coinvariants with a twisted vacuum insertion (Theorems 5.1 and 5.2).

This isomorphism Φ is constructed for fixed G-bundles P (and locally fixed Hecke parameters), relying on a local trivialization and specific coordinate changes. The authors then ask whether this construction can be extended when the underlying G-bundle is allowed to vary, i.e., whether Φ can be defined over a substack of the moduli of G-bundles with additional structures, thereby globalizing the coordinate-transformation description beyond the fixed-bundle setting.