The Kac-Moody group conjecture for the monoid of integrable representations

Let \text{\bf g} be a symmetrizable Kac-Moody algebra, let $G$ be the (minimal) Kac-Moody group, and let $M_{drr}$ be the monoid associated to the category of integrable \text{\bf g}-modules whose integrable duals are point separating, together with its category of integrable duals. Kac-Moody group conjecture. The group $G$ identifies with the monoid $M_{drr}$, and the Kac-Moody algebra \text{\bf g} is the Lie algebra of $M_{drr}$. This conjecture concerns the Tannaka reconstruction of the integrable representation category; the preceding discussion establishes the relevant faithfulness assumptions, but the identification of the reconstructed monoid and its Lie algebra is left as a conjecture.