The g-vector formula from a single F-polynomial

Fix $B^0,t_0,\ell,t$ such that $F_{\ell;t}^{B^0;t_0}\neq1$, and write $\mathbf{g}_{\ell;t}^{B^0;t_0}=(g_1,\dots,g_n)$. Let $\operatorname{Trop}(u_1,\dots,u_n)$ denote the tropical semifield specialization. Single-F-polynomial g-vector conjecture.

$$u_1^{g_1}\cdots u_n^{g_n}=\frac{F_{\ell;t}^{B^0;t_0}|_{\operatorname{Trop}(u_1,\dots,u_n)}(u_1^{-1},\dots,u_n^{-1})}{F_{\ell;t}^{B^0;t_0}|_{\operatorname{Trop}(u_1,\dots,u_n)}\left(\prod_i u_i^{b^0_{i1}},\dots,\prod_i u_i^{b^0_{in}}\right)}.$$

The formula would recover the $\mathbf{g}$-vector from one $F$-polynomial alone. It is proved on the bipartite belt and remains open in general.