Universal positivity conjecture for quantum greedy elements when one exchange parameter divides the other

Let $b,c$ be positive integers, and let $X[a_1,a_2]$ denote the quantum greedy element of the rank $2$ quantum cluster algebra $\mathcal{A}_v(b,c)$ associated with $(a_1,a_2)\in\mathbb{Z}^2$. An element is universally positive if its Laurent expansion has positive coefficients in every cluster. Universal positivity conjecture. If $b\mid c$ or $c\mid b$, then all quantum greedy elements $X[a_1,a_2]$ are universally positive, and hence indecomposable.