The extension conjecture for parahoric torsors

Let $E$ be a $p$-adic field with perfect residue field, let $C$ be a perfect non-archimedean field of characteristic $p$ with an open and bounded valuation subring $C^+$, and set $A_E=W(C^+)\widehat\otimes_{W(k)}O_E$, $X=\operatorname{Spec}(A_E)$, and $U=X\setminus V(\pi,[\varpi])$. Let $G$ be a reductive group over $E$ and let ${\mathcal G}$ be a parahoric model of $G$ over $O_E$. Extension conjecture. Every ${\mathcal G}$-torsor over $U$ extends to a ${\mathcal G}$-torsor over $X$. The claim is a parahoric analogue of extension and triviality results for vector bundles; the source gives no resolution in the stated generality.