Birationality conjecture for minimal bipartite graphs

Let $G$ be a minimal bipartite graph on a torus whose Newton polygon has $d+2$ vertices, and let $\mathcal F \colon \mathcal B \to \mathcal S$ be the map from the space of white data to the spectral data. Birationality conjecture. For generic white data in dimension $d$, the map

$$\mathcal F \colon \mathcal B \to \mathcal S$$

is birational. Experimental evidence suggests that local moves may raise the degree of any chosen white vertex to the number of vertices of the Newton polygon, which would yield birationality in this setting; the conjecture remains open in general.