  The $B_{\mathrm{dR}}^+$-affine Grassmannian was introduced by Scholze in the
context of the geometric local Langlands program in mixed characteristic and is
the Fargues-Fontaine curve analogue of the equal characteristic
Beilinson-Drinfeld affine Grassmannian. For a reductive group $G$, it is
defined as the \'{e}tale (equivalently, $v$-) sheafification of the presheaf
quotient $LG/L^+G$ of the $B_{\mathrm{dR}}$-loop group $LG$ by the
$B_{\mathrm{dR}}^+$-loop subgroup $L^+G$. We combine algebraization and
approximation techniques with the known cases of the Grothendieck-Serre
conjecture to show that the analytic topology suffices for this sheafification,
more precisely, that the $B_{\mathrm{dR}}^+$-affine Grassmannian agrees with
the analytic sheafification of the aforementioned presheaf quotient $LG/L^+G$.
