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

The $B_{\mathrm{d}\mathrm{R}}^{+}$-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 étale (equivalently, $v$-) sheafification of the presheaf quotient $LG\mathrm{/}{L}^{+}G$ of the $B_{\mathrm{d}\mathrm{R}}$-loop group $LG$ by the $B_{\mathrm{d}\mathrm{R}}^{+}$-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 $B_{\mathrm{d}\mathrm{R}}^{+}$-affine Grassmannian agrees with the analytic sheafification of the aforementioned presheaf quotient $LG\mathrm{/}{L}^{+}G$.