The mutation-equivariance conjecture for g-vectors

Let $\mathbb{T}_n$ be the $n$-regular tree indexing seeds, let $t_0\stackrel{k}{\text{---\!---}}t_1$ be an edge, and let $B_0$ and $B_1$ be exchange matrices with $B_1=\mu_k(B_0)$. For any $t\in\mathbb{T}_n$ and $i\in[n]$, write $\mathbf{g}_{i;t}^{B_0;t_0}$ and $\mathbf{g}_{i;t}^{B_1;t_1}$ for the corresponding $\mathbf{g}$-vectors, and let $\eta^{B_0^T}_k$ denote the associated piecewise-linear transformation. The mutation-equivariance conjecture. The vectors satisfy

$$\mathbf{g}_{i;t}^{B_1;t_1}=\eta^{B_0^T}_k\left(\mathbf{g}_{i;t}^{B_0;t_0}\right),$$

where $B_0^T$ is the transpose of $B_0$. This is a weak version of the cited conjecture on $\mathbf{g}$-vectors and is used to prove the formula for $\mathbf{g}$-vectors associated with tagged arcs on surfaces; its resolution status is not specified in the supplied text.