The relative Langlands dual and derived Satake equivalence for symmetric spaces

Let $G$ be a reductive group with involution $\theta$, let $X_\theta=G^{\operatorname{inv}\circ\theta}=\{g\in G\mid\theta(g)=g^{-1}\}$ with its $\theta$-twisted conjugation action, and let $G^L=G\rtimes_\theta\mu_2$. Let $\check G$, $\check X_{\check\theta}(F)$, $\operatorname{Bun}_{\check G_{\mathbb R}}(\widetilde{\mathbb P}^1_{\mathbb R})$, and the functors $\Upsilon_{\check\theta}$ and $\Upsilon_{\mathbb R}$ be as in the preceding constructions. There are horizontal equivalences in the diagram

\xymatrix{D^b_{\check G(\mathcal O)}(\check X_{\check\theta}(F))\ar[r]^{\Upsilon_{\check\theta}\ \ }_{\simeq\ \ \ }\ar[d]_{\simeq}^{~}&\operatorname{Perf}^G(T^*X_\theta[2])\ar[d]_{\simeq}^{}\\ D_!(\operatorname{Bun}_{\check G_{\mathbb R}}(\widetilde{\mathbb P}^1_{\mathbb R}))\ar[r]^{\Upsilon_{\mathbb R}}_{\simeq}&\operatorname{Coh}(\operatorname{Loc}_{G^L}(\widetilde{\mathbb P}^1_{\mathbb R}))}

This would identify the dual-side equivariant derived category and the real affine Grassmannian category with the two sides of the relative Koszul-duality equivalence, providing a relative Langlands dual realization of $T^*(G/K)$ and a derived Satake correspondence. The supplied text does not indicate whether the proposed equivalences have been established, so their resolution remains open.