The g-vector parametrization conjecture for cluster monomials

Let $\mathcal{A}$ be a cluster algebra with initial seed at $t_0$, and let $\mathbf{g}_{\mathbf{a};t}^{B^0;t_0}$ be the $\mathbf{g}$-vector of a cluster monomial $x_{\mathbf{a};t}$. g-vector parametrization conjecture. (1) Distinct cluster monomials have distinct $\mathbf{g}$-vectors with respect to a fixed initial seed. (2) For every $t\in\mathbb{T}_n$, the vectors $\mathbf{g}_{1;t}^{B^0;t_0},\dots,\mathbf{g}_{n;t}^{B^0;t_0}$ form a $\mathbb{Z}$-basis of $\mathbb{Z}^n$. These properties would provide an integral parametrization of cluster monomials by $\mathbf{g}$-vectors. The paper gives finite-type and bipartite evidence, but the general conjecture remains open.