LLRZ's support conjecture for triangular bases of rank-2 cluster algebras

For positive integers $b,c$, define the coefficient-free rank-2 quantum cluster algebra $\mathcal{A}_{\mathbf v}(b,c)$. Let

$$\Phi^{im}_+=\{(a_1,a_2)\in\mathbb{Z}^2_{>0}: ca_1^2-bca_1a_2+ba_2^2\le0\}$$

be the set of positive imaginary roots. For $(a_1,a_2)\in\Phi^{im}_+$, let $C[a_1,a_2]$ be a triangular basis element, written as

$$C[a_1,a_2]=\sum_{p,q}e(p,q)X^{(bp-a_1,cq-a_2)}.$$

For integers $0\le p\le a_2$ and $0\le q\le a_1$, define

$$D(p,q)=ca_1q+ba_2p-bp^2-bcpq-cq^2.$$

LLRZ's support conjecture. The coefficient $e(p,q)$ of $C[a_1,a_2]$ is nonzero if and only if $D(p,q)\ge0$.

This gives an explicit description of the support of triangular basis elements for positive imaginary roots in rank-2 quantum cluster algebras. The paper proves the broader support conjecture for all skew-symmetric rank-2 cluster algebras, while the cited formulation is attributed to Lee, Rupel, Zelevinsky and the author.