The geometric endomorphism conjecture for the unramified Hecke module

Let $\mathbf{G}$ be a reductive group over the nonarchimedean field $k$, let $\mathbf{X}$ be a spherical $\mathbf{G}$-variety, let $K$ be a maximal compact subgroup of $G=\mathbf{G}(k)$, and let $A_X^*$ and $W_X$ denote the dual torus and little Weyl group associated with $\mathbf{X}$. Write $\mathcal H(G,K)$ for the Hecke algebra and let $S_\chi^{\mathring X}$ be the family of morphisms associated with the open $B$-orbit. Call an endomorphism of $(C_c^\infty(X))^K$ geometric if it preserves, up to a rational multiple, the family of morphisms $S_\chi^{\mathring X}$. Geometric endomorphism conjecture. There is a canonical isomorphism

$$\left(\operatorname{End}_{\mathcal H(G,K)}C_c^\infty(X)^K\right)^{\operatorname{geom}}\simeq \mathbb{C}[\delta^{-\frac{1}{2}}A_X^*]^{W_X}$$

such that the diagram relating $\mathcal H(G,K)$, the geometric endomorphism algebra, $\mathbb{C}[A^*]^W$, and $\mathbb{C}[\delta^{-\frac{1}{2}}A_X^*]^{W_X}$ commutes. This conjecture proposes an intrinsic commutative endomorphism algebra for the unramified Hecke module, analogous to the algebra of invariant differential operators on a spherical variety; the source does not state a resolution or identify cases in which it is known.