Fomin–Zelevinsky's tropical formula for g-vectors

Let $F_{i;v}$ be the $F$-polynomial associated with the cluster variable $x_{i;v}$, and let $\mathbf{g}(x_{i;v})=(g_1,\dots,g_m)$ be its g-vector. Write $u_1,\dots,u_m$ for tropical semifield generators and $b_{sj}$ for the entries of the exchange matrix. Fomin–Zelevinsky's tropical g-vector conjecture. Suppose that $F_{i;v}$ is not identically equal to $1$. Then

$$\prod_{i=1}^m u_i^{\tilde{g}_i}=\dfrac{F_{i;v}|_{\operatorname{Trop}(u_1,\dots,u_m)}(u_1^{-1},\dots,u_n^{-1})}{F_{i;v}|_{\operatorname{Trop}(u_1,\dots,u_m)}(\prod_{s=1}^m u_s^{b_{s1}},\dots,\prod_{s=1}^m u_s^{b_{sn}})}.$$

Here specialization to $\operatorname{Trop}(u_1,\dots,u_m)$ means evaluation in the tropical semifield. The statement is attributed to Fomin and Zelevinsky and concerns recovering g-vectors from $F$-polynomials; its resolution is not established by the supplied text.