Fomin–Zelevinsky recurrence conjecture for g-vectors

Fix a regular tree $\mathbb T_n$. For vertices $t_1,t_2\in\mathbb T_n$ joined by an edge labeled $k$, write $t_1\overset{k}{--}t_2$, let $B^2=\mu_k(B^1)$, and for $a\in[1,n]$ and $t\in\mathbb T_n$ write

$$\mathbf g^{B^1;t_1}_{a;t}=(g_1^{t_1},\ldots,g_n^{t_1}),\qquad \mathbf g^{B^2;t_2}_{a;t}=(g_1^{t_2},\ldots,g_n^{t_2}).$$

Here $[r]_+=\max(r,0)$ and $\min$ denotes the minimum function. Fomin–Zelevinsky's g-vector recurrence conjecture. Under these assumptions,

$$g_i^{t_2}=\begin{cases}-g_k^{t_1}&\text{if }i=k,\g_i^{t_1}+[b_{ik}^{t_1}]_+g_k^{t_1}-b_{ik}^{t_1}\min(g_k^{t_1},0)&\text{if }i\ne k.\end{cases}$$

This is presented as a conjectural change-of-principal-seed relation for g-vectors; the supplied text does not establish its general status, although the paper proves related recurrence results in the acyclic sign-skew-symmetric setting.