Upper cluster structure on Kac--Moody Richardson varieties

We show coordinate rings of open Richardson varieties are upper cluster algebras for any symmetrizable Kac–Moody type. We further show the coordinate rings of (generalized) open Richardson varieties on the twisted product of flag varieties are upper cluster algebras for any symmetrizable Kac–Moody type. This includes, as special cases, reduced double Bruhat cells, Bott-Samelson varieties, braid varieties. Our results generalize various results by Casals–Gorsky–Gorsky–Le–Shen–Simental and Galashin–Lam–Sherman-Bennett–Speyer in finite types.