  We prove that torsors under parahoric group schemes on the punctured spectrum
of Fontaine's ring $A_{\mathrm{inf}}$, extend to the whole spectrum. Using
descent we can extend a similar result for the ring $\mathfrak{S}$ of Kisin and
Pappas to full generality. Moreover, we treat similarly the case of equal
characteristic. As applications we extend results of Ivanov on exactness of the
loop functor and present the construction of a canonical specialization map
from the $B^+_{\mathrm{dR}}$-affine Grassmannian to the Witt vector affine flag
variety.
