Equality of KR-polynomials and canonical basis elements

For $(i,p)\in{\widehat{\triangle}}_0$ and $k\in\mathbb{N}$, define the KR-monomial

$$m_{k,p}^{(i)}:=X_{i,p}X_{i,p+2}\cdots X_{i,p+2k-2}.$$

For a KR-monomial $m$, let $F_q(m)$ be its associated KR-polynomial, and let $L_q(m)$ be the corresponding canonical-basis element. Equality conjecture for KR-polynomials. For every KR-polynomial $F_q(m)$, one has

$$F_q(m)=L_q(m).$$

The conjecture is proved for simply-laced type by Nakajima; its validity for the general setting considered in the paper remains open.