Geometric Langlands duality compatibility with Serre duality

Let $G$ be a reductive group, let $X$ be the underlying smooth projective curve, and let $\bun_G$ be the moduli stack of $G$-bundles on $X$. Write $\bG$ for the Langlands dual group, $\fLocSys_{\bG}$ for the derived stack of $\bG$-local systems on $X$, and $\bDmod(\bun_G)$ and $\bIndCoh_{ ilpglob}(\fLocSys_{\bG})$ for the corresponding DG categories. Suppose that the geometric Langlands equivalence is

$$\BL_G:\bDmod(\bun_G)\to \bIndCoh_{\nilpglob}(\fLocSys_{\bG}).$$

Let $\bD^{\mathrm{Serre}}_{\fLocSys_{\bG}}$ be the Serre duality equivalence on the spectral category, let $\psId_{\bun_G,!}$ be the pseudo-identity functor, and let $\tau$ be the automorphism induced by the Cartan involution of $G$. Geometric Langlands duality conjecture. The diagram

$$\begin{CD} \bDmod(\bun_G)^\vee @>{((\BL_G)^\vee)^{-1}}>> (\bIndCoh_{\nilpglob}(\fLocSys_{\bG}))^\vee \\ @V{\psId_{\bun_G,!}}VV @VV{\bD^{\mathrm{Serre}}_{\fLocSys_{\bG}}}V \\ \bDmod(\bun_G) @>{\BL_G}>> \bIndCoh_{\nilpglob}(\fLocSys_{\bG}) \\ @. @VV{\tau}V \end{CD}$$

commutes up to a cohomological shift, with the indicated Cartan-involution automorphism. This is a motivational enhancement of the categorical geometric Langlands conjecture; the source gives no resolution, so the compatibility remains open.