Foldings of KLR algebras

Let ${\mathbf{U}}_q^{-}$ be the negative half of the quantum group associated to a Kac-Moody algebra $\mathfrak{g}$, and ${\underset{‾}{\mathbf{U}}}_q^{-}$ the quantum group obtained by a folding of $\mathfrak{g}$. Let $\mathbf{A}=\mathbf{Z}[q,q^{-1}]$. McNamara showed that ${\underset{‾}{\mathbf{U}}}_q^{-}$ is categorified over a certain extenion ring $\tilde{\mathbf{A}}$ of $\mathbf{A}$, by uing the folding theory of KLR algebras. He posed a question whether $\tilde{\mathbf{A}}$ coincides with $\mathbf{A}$ or not. In this paper, we give an affirmative answer for this problem.