Strongly tempered inverse Satake conjecture

Let $X$ be a strongly tempered spherical variety, so that its dual group satisfies $\check{G}_X=\check{G}$. Let $\mathbf{A}=\mathbf{A}_X$, let $\delta$ denote the relevant modular character, let $\Lambda^+$ be the dominant coweight cone, and let $\mathcal{L}$, $\hat{\mathcal{L}}$, and $\Phi_0$ be the operators and function appearing in the inverse Satake transform. Let $\Theta_X^+$ be the multiset of weights of $\check{\mathbf{A}}$ denoted by $\mathfrak{B}^+$ in Sakellaridis and Wang's work. Strongly tempered inverse Satake conjecture. One has

$$\delta^{\frac{1}{2}}\cdot(\mathcal{L}\cdot\Phi_0)=\hat{\mathcal{L}}\cdot\frac{\prod_{\check{\gamma}\in\check{\Phi}^+}(1-e^{\check{\gamma}})}{\prod_{\check{\theta}\in\Theta_X^+}(1-q^{-\frac{1}{2}}e^{\check{\theta}})}\bigg|_{\Lambda^+}$$

and all weights of $\Theta_X^+$ are minuscule as coweights of $\mathbf{G}$. In equal characteristic, the conjecture without the final minuscule-weight assertion is known by the main theorem of Sakellaridis and Wang; the final assertion is the remaining conjectural part.