Quantum positivity and canonical-basis equality for Kirillov–Reshetikhin monomials

Let $m^{(i)}[p,s]$ be a Kirillov–Reshetikhin monomial, and let $F_q(m^{(i)}[p,s])$ and $L_q(m^{(i)}[p,s])$ denote the associated KR-polynomial and canonical-basis element. KR-polynomial conjecture. For every KR-monomial $m^{(i)}[p,s]$, the element $F_q(m^{(i)}[p,s])$ is quantum positive, meaning that every coefficient belongs to

$$\mathbb{N}_0[q^{\pm 1/2}].$$

Moreover,

$$L_q(m^{(i)}[p,s])=F_q(m^{(i)}[p,s]).$$

The conjecture links the quantum cluster-algebra construction of KR-polynomials with the canonical basis. The source reports that the positivity assertion is proved by Nakajima in symmetric cases, while the general statement remains open.