Linear independence conjecture for g-vectors

Let $B$ be an exchange matrix and let $t_1,\ldots,t_m$ be vertices of the $n$-regular tree $\mathbb T_n$. Let $i_1,\ldots,i_m\in[n]$ and let $c_1,\ldots,c_m\in\mathbb Z$. For every vertex $t$, consider the corresponding $\boldsymbol{g}$-vectors $\boldsymbol{g}_{i_r;t}^{B_t;t}$, with the indices and vertices as in the source. Linear independence conjecture. If

$$\sum_{r=1}^m c_r\boldsymbol{g}_{i_r;t}^{B_t;t_r}=\mathbf 0$$

for every vertex $t$ of $\mathbb T_n$, then $c_j=0$ for every $j=1,\ldots,m$. The statement is proposed as an independence property of $\boldsymbol{g}$-vectors, and no resolution is given in the supplied text.