Parabolic compatibility of quantum Langlands for the Siegel parabolic

Let $G=\operatorname{Sp}(M_n)$, let $H=\operatorname{SO}_{2n+1}$, and let $P\subset G$ be the Siegel parabolic with Levi quotient $\operatorname{GL}_n$. Let $\operatorname{Eis}:\operatorname{D}(\operatorname{Bun}_n)\to\operatorname{D}_-(\operatorname{\widetilde{Bun}}_G)$ and $\operatorname{Eis}_{\operatorname{GL}_n}^H:\operatorname{D}(\operatorname{Bun}_n)\to\operatorname{D}(\operatorname{Bun}_H)$ be the corresponding geometric Eisenstein series functors.

Parabolic compatibility conjecture. There is an automorphism $\delta:\operatorname{Bun}_n\widetilde\to\operatorname{Bun}_n$ such that the diagram

$$\begin{array}{ccc} \operatorname{D}(\operatorname{Bun}_n)&\xrightarrow{\operatorname{Eis}}&\operatorname{D}_-(\operatorname{\widetilde{Bun}}_G)\\\\ \uparrow^{\delta^*}&&\uparrow^{QL}\\\\ \operatorname{D}(\operatorname{Bun}_n)&\xrightarrow{\operatorname{Eis}_{\operatorname{GL}_n}^H}&\operatorname{D}(\operatorname{Bun}_H) \end{array}$$

\nis 2-commutative. If $n=1$, then $\delta({\cal B})={\cal B}\otimes\Omega^{1/2}$ for some square root $\Omega^{1/2}$ of $\Omega$.

This expresses compatibility of quantum Langlands with parabolic induction. The source gives the assertion as conjectural and does not state a resolution.