The quantum-number positivity conjecture for quantum dilogarithm factorisations

Let $Q$ be the quiver with two vertices $0,1$, no loops, and $k\in\mathbb{Z}_{\geq 1}$ arrows from $0$ to $1$. For $n\in\mathbb{Z}_{\geq 1}$, define the signed quantum numbers

$$[\pm n]_q=(-1)^{n+1}\frac{q^{n/2}-q^{-n/2}}{q^{1/2}-q^{-1/2}}.$$

Using the quantum torus variables with multiplication $T^{\mathbf d}T^{\mathbf d'}=(-q^{1/2})^{-\langle\mathbf d',\mathbf d”\rangle_Q}T^{\mathbf d+\mathbf d'}$, write

$$\mathbb{E}([\pm n_1]_qT^{(0,1)})\mathbb{E}([\pm n_2]_qT^{(1,0)})=\prod_{\infty\xrightarrow{\theta}-\infty}\mathbb{E}(g_\theta),$$

where $g_\theta=\sum_{d_1/d_2=\theta}g_{d_1,d_2}(q^{1/2})T^{(d_1,d_2)}$. Quantum-number positivity conjecture. For every $n_1,n_2,k\in\mathbb{Z}_{\geq 1}$, each coefficient $g_{d_1,d_2}(q^{1/2})$ is a sum of elements of $\{[\pm n]_q\mid n\geq1\}$. This is a proposed Lefschetz-type positivity property for quantum theta-function factorisations.