Geometric Langlands compatibility with Eisenstein series

Let $G$ be a reductive group, let $P\subset G$ be a standard parabolic with Levi subgroup $M$, and let $\bP\subset \bG$ be the corresponding parabolic in the Langlands dual group with Levi subgroup $\bM$. Write $\bun_G$, $\bun_M$ for the moduli stacks of bundles and $\fLocSys_{\bG}$, $\fLocSys_{\bM}$ for the derived stacks of local systems. Let

$$\Eis_!:\bDmod(\bun_M)\to\bDmod(\bun_G),\qquad \Eis_{\mathrm{spec}}:\bIndCoh_{\nilpglob}(\fLocSys_{\bM})\to\bIndCoh_{\nilpglob}(\fLocSys_{\bG})$$

be the automorphic and spectral Eisenstein functors. Eisenstein-series compatibility conjecture. The diagram

$$\begin{CD} \bDmod(\bun_G) @>{\BL_G}>> \bIndCoh_{\nilpglob}(\fLocSys_{\bG}) \\ @A{\Eis_!}AA @AA{\Eis_{\mathrm{spec}}}A \\ \bDmod(\bun_M) @>{\BL_M}>> \bIndCoh_{\nilpglob}(\fLocSys_{\bM}) \end{CD}$$

commutes up to an automorphism of $\bIndCoh_{\nilpglob}(\fLocSys_{\bM})$ given by tensoring with a certain canonically defined graded line bundle on $\fLocSys_{\bM}$. This is a compatibility requirement in the categorical geometric Langlands program; the source provides no resolution, so it remains open.