Strong positivity conjecture for cluster algebras

Let $\mathcal{A}$ be a cluster algebra, and let a cluster monomial mean a monomial in cluster variables all belonging to the same cluster. Strong positivity conjecture. Any cluster algebra has an additive basis $\mathbb{B}$ which includes the cluster monomials and has nonnegative structure constants: when the product of any two elements of $\mathbb{B}$ is written in terms of $\mathbb{B}$, the coefficients are positive. The source states that this conjecture implies both the positivity conjecture and the linear independence conjecture for cluster monomials, but gives no general resolution.