Compatibility of the enhanced spectral Eisenstein functors with the global self-duality

Let $G$ be a reductive group, let $P^-$ be an opposite parabolic with Levi subgroup $M$, and let $\CZ$ be the auxiliary coefficient category. Consider the equivalence

$$\Theta_{\on{I}(\cG,\cP^-)^{\on{spec,glob}}}:\IndCoh(\LS_\cM)^{-,\on{enh}_{\on{co}}}_\CZ\longrightarrow \IndCoh(\LS_\cM)^{-,\on{enh}}_\CZ$$

and the enhanced spectral Eisenstein functors appearing in the diagram below.

Compatibility conjecture. The following diagram of functors commutes:

$$\CD \IndCoh(\LS_\cM)^{-,\on{enh}_{\on{co}}}_\CZ @>{\Theta_{\on{I}(\cG,\cP^-)^{\on{spec,glob}}}}>> \IndCoh(\LS_\cM)^{-,\on{enh}}_\CZ \\ @V{\Eis_\CZ^{-,\on{spec},\on{enh}_{\on{co}}}}VV @VV{\Eis_\CZ^{-,\on{spec},\on{enh}}}V \\ \IndCoh(\LS_\cG)\otimes \Dmod(\CZ) @>>{\on{id}}> \IndCoh(\LS_\cG)\otimes \Dmod(\CZ), \endCD$$

in a way compatible with the $\Sph^{\on{spec}}_{\cG,\CZ}$-actions.

The conjecture expresses compatibility between the proposed global self-duality and enhanced spectral Eisenstein series. The source gives a local and categorical formulation, but no resolution of this compatibility statement.