Kottwitz-style component conjecture for parahoric bundle moduli

Assume that $\operatorname{char}(k)\neq 2$, that $\pi:\widetilde X\rightarrow X$ is a possibly ramified double cover with involution $\sigma$, and set $\mathcal G={\rm SU}_n(\widetilde X/X)$. Let $\mathcal M_{\mathcal G/X}$ be the moduli stack of $\mathcal G$-torsors, and let $\bar\eta$ and $\eta$ denote geometric and generic points of $X$. Denote by $\pi_1(\mathcal G_{\bar\eta})$ the algebraic fundamental group in the sense of Borovoi, and write $\Gamma=\operatorname{Gal}(\bar\eta/\eta)$. Kottwitz-style component conjecture. The connected components satisfy

$$\pi_0(\mathcal M_{\mathcal G/X})=\pi_1(\mathcal G_{\bar\eta})_{\Gamma}.$$

Here the right-hand side denotes the coinvariants under $\Gamma$. In particular, if $\mathcal G_{\bar\eta}$ is semisimple and simply connected, the moduli stack should be connected. The statement is known in the constant case over $\mathbf C$ by topological uniformization, but is open in the stated generality.