Formality conjecture for de-equivariantized Ext algebras in relative Langlands duality

Let $G$ be a reductive group, let $X$ be a spherical variety, and let $D^{}(L^+G\backslash LX)$ be the derived relative Satake category of $L^+G$-equivariant complexes on $LX$. Let $\mathrm{e}_{L^+X}$ be the constant sheaf on the closed orbit $LX_0=L^+X$, and let $\IC_{\cO(\check G)}$ be the image of the regular representation $\cO(\check G)$ under the abelian Satake equivalence. Define the de-equivariantized Ext algebra

$$A_X:=R\Hom_{D(L^+G\backslash LX)}(\mathrm{e}_{L^+X},\IC_{\cO(\check G)}\star\mathrm{e}_{L^+X}).$$

Formality conjecture. The dg-algebra $A_X$ is formal: it is quasi-isomorphic to the graded algebra

$$A_X\cong\operatorname{Ext}^\bullet_{D(L^+G\backslash LX)}(\mathrm{e}_{L^+X},\IC_{\cO(\check G)}\star\mathrm{e}_{L^+X})$$

with trivial differential. This is a conjecture in relative Langlands duality for spherical varieties; the supplied text gives no evidence of a resolution.