Heine's conjecture for volume polynomials of three lattice polygons

Let $M$ be a $3\times 3$ symmetric matrix with non-negative integer entries and let

$$F(x,y,z)=\left(x \,\, y\,\, z\right)M\left(\begin{matrix}x \\ y\\ z\end{matrix}\right)$$

be the corresponding ternary form. Assume that $\det(M)\geq 0$ and $\det(M_I)\leq 0$, where $\det(M_I)$ is the diagonal $2\times 2$ minor indexed by $I\subset\{1, 2, 3\}$. Heine's conjecture. There exist lattice polygons $P,Q,R\subset\mathbb R^2$ such that

$$F(x,y,z)=\operatorname{Vol}_2(xP+yQ+zR).$$

This is a discrete analogue of Heine's description of volume polynomials of triples of planar convex bodies. The corresponding realization problem is open beyond the cases established in the paper, and the general Heine–Shephard problem for volume polynomials remains open.