Normalization conjecture for the Kac–Moody unipotent automorphism group

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

$$\widehat{{U}}=\widehat{{U}}_1=\left\{\varphi\in\operatorname{Aut}(\widehat{\mathfrak g})\mid \varphi(y)\in y+\widehat{\mathfrak g}_{k+1}\text{ whenever }y\in\mathfrak g_k\right\}.$$

Define

$$\mathcal H=\left\{\varphi\in\operatorname{Aut}(\widehat{\mathfrak g})\mid \varphi(y)\in\mathfrak g_k\text{ whenever }y\in\mathfrak g_k\right\}.$$

Normalization conjecture. The group $\mathcal H$ normalizes $\widehat{{U}}$. This is one of the conjectures proposed for extensions of Scott's definition in the Kac–Moody setting; its resolution concerns the interaction between grading-preserving automorphisms and the unipotent automorphism group.