The g-vector transition conjecture

Let $t_0\xrightarrow{k}t_1$ be adjacent vertices of $\mathbb{T}_n$, let $B^1=\mu_k(B^0)$, and fix $t\in\mathbb{T}_n$ and $\mathbf{a}\in\mathbb{Z}_{\geq0}^n$. Write $\mathbf{g}_{\mathbf{a};t}^{B^0;t_0}=(g_1,\dots,g_n)$ and $\mathbf{g}_{\mathbf{a};t}^{B^1;t_1}=(g'_1,\dots,g'_n)$. g-vector transition conjecture.

$$g'_j=\begin{cases}-g_k,&j=k,\\ g_j+[b^0_{jk}]_+g_k-b^0_{jk}\min(g_k,0),&j\ne k.\end{cases}$$

This is a conjectural change-of-initial-seed rule for $\mathbf{g}$-vectors, related in the paper to tropical $Y$-patterns and Langlands duality. It remains open in general.