Kashiwara–Kim–Oh–Park's monoidal categorification conjecture for open Richardson varieties

Let $W$ be the Weyl group of a Dynkin diagram, and let $w,v,u\in W$ satisfy

$$w=vu,\qquad \ell(w)=\ell(v)+\ell(u).$$

Let $\mathscr{C}_{w,v}$ be the subcategory associated with $w$ and $v$, let $K_q(\mathscr{C}_{w,v})$ denote its quantum Grothendieck group, and let $A_q(\mathfrak{n}(u^{-1}))_{\mathbb{Z}[q,q^{-1}]}$ be the quantum cluster algebra associated with $\mathfrak{n}(u^{-1})$ over $\mathbb{Z}[q,q^{-1}]$. Kashiwara–Kim–Oh–Park's conjecture. The category $\mathscr{C}_{w,v}$ gives a monoidal categorification of the quantum cluster algebra

$$A_{w,v}=K_q(\mathscr{C}_{w,v}),$$

induced from the monoidal categorification of $A_q(\mathfrak{n}(u^{-1}))_{\mathbb{Z}[q,q^{-1}]}$ via the reflection functors. The conjecture concerns extending monoidal categorification from quantum coordinate rings associated with unipotent subgroups to the quantum cluster algebras of open Richardson varieties; the cited paper proposes it for Dynkin type, while the present paper's abstract says that its results confirm the conjecture.