Gabber–Kac simplicity conjecture for complete Kac–Moody groups

Let $A$ be a generalized Cartan matrix and let $k$ be a field of characteristic zero or bigger than $M_A:=\max_{i\neq j}|a_{ij}|$. The group $\mathfrak{G}_A^{pma}(k)$ is GK-simple, meaning that every normal subgroup of $\mathfrak{G}_A^{pma}(k)$ contained in $\mathfrak{U}_A^{ma+}(k)$ is trivial.

Gabber–Kac simplicity conjecture. The group $\mathfrak{G}_A^{pma}(k)$ is GK-simple.

This conjecture concerns the injectivity of the natural morphism from the Mathieu–Rousseau completion to the complete representation-theoretic completion. It is motivated by the Gabber–Kac theorem for symmetrisable Kac–Moody algebras; the corresponding group-theoretic assertion is open in the stated generality.