Decomposition conjecture for the completed Kac–Moody automorphism group

Let $\widehat{\mathfrak g}$ be the completed Kac–Moody algebra, with graded pieces $\mathfrak g_k$. Define

$$\widehat{{U}}_0=\left\{\varphi\in\operatorname{Aut}(\widehat{\mathfrak g})\mid \varphi(y)\in\widehat{\mathfrak g}_k\text{ whenever }y\in\mathfrak g_k\right\},$$

and let $\widehat{{U}}=\widehat{{U}}_1$ and $\mathcal H$ be as above. Decomposition conjecture. The group $\widehat{{U}}_0$ admits the decomposition

$$\widehat{{U}}_0=\widehat{{U}}H=H\widehat{{U}}.$$

This is proposed as part of the paper's extension of Scott's definition to Kac–Moody groups. It asserts that every element of $\widehat{{U}}_0$ can be expressed using the unipotent subgroup and the grading-preserving subgroup, although the source supplies no evidence that the conjecture has been resolved.