q-commutativity conjecture for KR-polynomials

Let $(m^{(i)}_{k,r},m^{(j)}_{l,t})$ be a pair of KR-monomials, and let $F_q(\underline{m}^{(i)}_{k,r})$ and $F_q(\underline{m}^{(j)}_{l,t})$ be the associated KR-polynomials. Let $\mathsf{h}$ be the relevant Coxeter number and $\widetilde{\mathsf{b}}_{i,j}$ the indicated coefficient function. q-commutativity conjecture for KR-polynomials. The two KR-polynomials $q$-commute unless there exist $1\leqslant u\leqslant\mathsf{h}$ and $0\leqslant s\leqslant\min(k,l)-1$ satisfying

$$|k+r-l-t|=u+|k-l|+2s$$

and

$$\widetilde{\mathsf{b}}_{i,j}(u-1)\ne0.$$

The paper notes that this conjecture is proved in the cited work for finite $AD$-type Lie algebras, while the general claim remains unresolved in the supplied context.