g-vector compatibility criterion for common clusters

Let $B$ be an exchange matrix with principal coefficients and initial vertex $t_0$. Let $x_{i;t'}^{B;t_0}$ and $x_{j;t”}^{B;t_0}$ be cluster variables. For any vertex $t$ of the $n$-regular tree $\mathbb T_n$, let $B_t$ be the exchange matrix at $t$, and let the corresponding $\boldsymbol{g}$-vectors be denoted by $\boldsymbol{g}_{i;t'}^{B_t;t}$ and $\boldsymbol{g}_{j;t”}^{B_t;t}$. $\boldsymbol{g}$-vector compatibility conjecture. The variables $x_{i;t'}^{B;t_0}$ and $x_{j;t”}^{B;t_0}$ are not contained in any common cluster if and only if there exist $t\in\mathbb T_n$ and $k\in[n]$ such that these two $\boldsymbol{g}$-vectors have strictly opposite signs in their $k$th entries. One implication is proved under the standard hypotheses; the converse remains conjectural.