Integer-basis conjecture for g-vectors

Let $B$ be an exchange matrix, let $t_0$ be a fixed initial vertex of the $n$-regular tree $\text{T}_n$, and let $t$ be any vertex. The vectors $\boldsymbol{g}_{i;t}^{B;t_0}$ are the $\boldsymbol{g}$-vectors of the cluster at $t$. Integer-basis conjecture. For every $t\in\text{T}_n$, the vectors

$$(\boldsymbol{g}_{i;t}^{B;t_0}:i\in[n])$$

form a $\mathbb Z$-basis of $\mathbb Z^n$. The source places this among the standard hypotheses and uses it in the discussion of positive bases; no resolution is supplied here.