Finiteness of $\mathbb{Q}$-Fano compactifications of semisimple group with K\"ahler-Einstein metrics

In this note, we give a way to classify $\mathbb{Q}$-Fano compactifications of a semisimple group $G$. We will prove that there are only finitely many such $\mathbb{Q}$-Fano $G$-compactifications, which admits (singular) K"ahler-Einstein metrics. As an application, this improves a former result in arXiv:2001.11320.