Conjectural Iwahori-equivariant orthosymplectic Satake equivalence

Let $V_0$ and $V_1$ be respectively orthogonal and symplectic vector spaces, let $\Ffl_1$ be the variety of complete self-orthogonal flags in $V_1$, and let $\Ffl_0$ be a chosen connected component of the variety of complete self-orthogonal flags in $V_0$. Define the shifted dg-scheme with trivial differential

$$\on{H}_\osp:=(V_0\otimes V_1)[1]\times\Ffl_0\times\Ffl_1.$$

For $A\in V_0\otimes V_1\cong\operatorname{Hom}(V_0,V_1)$, write $A^t\in\operatorname{Hom}(V_1,V_0)$ for its adjoint, and write $F_i=(F_i^{(1)}\subset F_i^{(2)}\subset\cdots\subset F_i^{(\dim V_i)}=V_i)$ for a flag. The orthosymplectic Steinberg scheme is

$$\on{St}_\osp=\{(A,F_0,F_1)\in\on{H}_\osp\mid A^tA(F^{(r)}_0)\subseteq F^{(r)}_0\ \text{and}\ AA^t(F^{(r)}_1)\subseteq F^{(r)}_1,\ \forall r\}.$$

Let $\bI_{N-1}\subset\SO(N-1,\bO)$ and $\bI_N\subset\SO(N,\bO)$ be Iwahori subgroups, let $\Fl_{\SO_N}:=\SO(N,\bF)/\bI_N$, and let $D^b_{\bI_{N-1}}(\Fl_{\SO_N})$ be the bounded $\bI_{N-1}$-equivariant constructible derived category. The conjectural Iwahori-equivariant orthosymplectic Satake equivalence. There exists an equivalence of triangulated categories

$$D^{\SO(V_0)\times\Sp(V_1)} \operatorname{Coh}(\on{St}_\osp)\cong D^b_{\bI_{N-1}}(\Fl_{\SO_N}).$$

This proposes an Iwahori-equivariant enhancement of the orthosymplectic Satake correspondence, relating coherent sheaves on the orthosymplectic Steinberg scheme to constructible sheaves on an affine flag variety. The source provides no resolution status for this equivalence.