Existence conjecture for cocompact lattices in the local pro-$p$ completion

Let $A$ be the rank-$2$ generalized Cartan matrix considered in the source, with $\max(a_{21},a_{12})\leq -2$ and without assuming $a_{21}=a_{12}$. Let $\widehat{G}$ denote the local pro-$p$ completion of the associated Kac–Moody group. A cocompact lattice is a discrete subgroup $\Gamma\leq\widehat{G}$ such that $\Gamma\backslash\widehat{G}$ is compact. The cocompact-lattice existence conjecture. The Kac–Moody group $\widehat{G}$ admits a cocompact lattice. The claim is posed because the preceding proof does not cover the nonsymmetric and $a_{12}=-1$ cases; its status is open in the source.