The analytic topology suffices for the $B_{\mathrm{dR}}^+$-Grassmannian

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 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 $LG/L^+G$.