The discrete Shephard conjecture for volume polynomials of two lattice polytopes

Let $F(x,y)=\sum_{i=0}^d{d\choose i}a_ix^iy^{d-i}$ be a degree $d$ form, where $a_0,\dots,a_d$ are non-negative integers satisfying

$$a_{i-k}a_{j+k}\leq a_ia_j$$

for all $0\leq i\leq j\leq d$ and $0\leq k\leq\min(i,d-j)$. The discrete Shephard conjecture. There exist lattice polytopes $P,Q\subset\mathbb R^d$ such that

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

This is the discrete counterpart of Shephard's characterization of volume polynomials for pairs of convex bodies, whose coefficients satisfy log-concavity inequalities. The statement is confirmed for $d=2$ by the theorem cited in the source, while the general case remains open.