Affineness conjecture for the geometric motivic group $K_{\mathbb{Z},n}$

Over $\operatorname{Spec}(\mathbb{Z})$, let $G^{\bullet}_{\mathbb{Z},n}$ be the affine derived group scheme associated with the mixed Tate category of $\mathcal{M}_{0,n}$, and let $K^{\bullet}_{\mathbb{Z},n}$ be the kernel of its structure morphism to $G^{\bullet}(\mathbb{Z})$. Affineness conjecture. The affine derived group scheme $K^{\bullet}_{\mathbb{Z},n}$ is an affine group scheme, namely

$$K^{\bullet}_{\mathbb{Z},n}=K_{\mathbb{Z},n}=\operatorname{Spec}(R_n),$$

where $R_n$ is a commutative Hopf algebra, not necessarily cocommutative, defined over $\mathbb{Z}$. This is one of the paper's open questions concerning a finer understanding of the derived groups $K_n$.