Uniformization conjecture for parahoric bundle moduli

Assume the setup of a parahoric group scheme $\mathcal G$ over a curve $X$ and let $\mathcal M_{\mathcal G/X}$ denote the moduli stack of $\mathcal G$-torsors. Let $x\in X(k)$, let $S$ be a base scheme, and let $\mathcal P$ be a $\mathcal G$-torsor over $X\times_k S$. Uniformization conjecture. If $\mathcal G_\eta$ is semisimple, then after an fppf base change $S'\rightarrow S$, the restriction of $\mathcal P\times_S S'$ to $(X\smallsetminus\{x\})\times S'$ is trivial. This asserts uniformization of every family of torsors away from one chosen point; it is known in the constant case by Drinfeld and Simpson and over a point by Harder, while the general parahoric case remains open.