Positivity conjecture for real canonical-basis elements

Let $m\in\mathcal{M}_+$, and let $L_q(m)$ be the corresponding canonical-basis element of the quantum Grothendieck ring. Call $L_q(m)$ real if, for every $k\in\mathbb{Z}_{\geqslant1}$, there exists $t\in\mathbb{Z}$ such that $L_q(m)^k=q^tL_q(m^k)$. Positivity conjecture for real canonical-basis elements. If $L_q(m)$ is real, then $L_q(m)$ has a quantum positive coefficient. This conjecture extends coefficient positivity from KR-polynomials to all real canonical-basis elements; the supplied passage does not state a resolution.