Representability conjecture for truncated affine Grassmannian double quotients

Suppose that $G$ is a reductive group split over $k$, with a split Borel pair $(B,T)$, and let

$$\mathcal{L}G=\bigsqcup_{\mu\in X_*(T)_{\mathrm{dom}}}\mathcal{C}^{\mu}$$

be its Cartan decomposition. For a dominant cocharacter $\mu$, let $\mathcal{C}^{\leq\mu}$ denote the closure of $\mathcal{C}^{\mu}$, and let $\mathcal{K}_1$ be the level-one congruence subgroup. Define the fpqc $k$-sheaf

$$\prescript{}{1}{\mathcal{C}_1^{\leq\mu}}:=\mathcal{K}_1\backslash\mathcal{C}^{\leq\mu}/\mathcal{K}_1.$$

Representability conjecture. The fpqc sheaf $\prescript{}{1}{\mathcal{C}_1^{\leq\mu}}$ is represented by a normal $k$-scheme. This predicts that the indicated truncated double quotient, which is presently introduced as an fpqc sheaf, admits a concrete geometric moduli interpretation as a normal scheme. The paper presents it as an open problem, and no resolution is supplied here.