Positivity conjecture for graph LP algebras

Let ${\mathcal A}_\Gamma$ be a graph LP algebra and let $t_{\mathcal S}$ be a seed. Write ${\mathcal L}(t_{\mathcal S})$ for the Laurent polynomial ring associated with this seed. Positivity conjecture.

1. All exchange polynomials $F_i$ of $t_{\mathcal S}$ are positive polynomials in ${\mathcal L}(t_{\mathcal S})$.
2. Any cluster variable of ${\mathcal A}_\Gamma$ has positive coefficients when written as a Laurent polynomial in ${\mathcal L}(t_{\mathcal S})$.

The second assertion is Laurent by the Laurent phenomenon, implies the first assertion, and is known for the initial seed; computations in the paper support the conjecture. A combinatorial interpretation of the coefficients is posed separately as an open problem.