Mirabolic Iwahori-equivariant Satake conjecture

Let $V_1$ and $V_2$ be $N$-dimensional vector spaces, let $\bI$ be an Iwahori subgroup, and let $\bFfl_i$ be the variety of complete flags in $V_i$. Define the dg-scheme with zero differential

$$\boldsymbol{H}_{\mathrm{mir}}:=\operatorname{Hom}(V_1,V_2)[1]\times\operatorname{Hom}(V_2,V_1)[1]\times\bFfl_1\times\bFfl_2.$$

Writing $A$ and $B$ for the corresponding odd linear maps, define the mirabolic Steinberg scheme

$$\boldsymbol{St}_{\mathrm{mir}}=\{(A,B,F_1,F_2)\in\boldsymbol{H}_{\mathrm{mir}}\mid AB(F^{(j)}_2)\subseteq F^{(j)}_2\ \text{and}\ BA(F^{(j)}_1)\subseteq F^{(j)}_1,\ \forall j\in[1,N]\}.$$

Let $\Fl=\bG_\bF/\bI$ and let $D_\bI(\Fl\times\bV)$ denote the $\bI$-equivariant constructible derived category. Mirabolic Iwahori-equivariant Satake conjecture. There exists an equivalence of triangulated categories

$$D^{\operatorname{GL}(V_1)\times\operatorname{GL}(V_2)}\operatorname{Coh}(\boldsymbol{St}_{\mathrm{mir}})\cong D_\bI(\Fl\times\bV).$$

This is the proposed mirabolic counterpart of Bezrukavnikov's equivalence between the Iwahori-equivariant constructible derived category of the affine flag variety and an equivariant coherent-sheaf category on a dg Steinberg variety. Its status is not resolved in the source.