Bøgvad's conjecture on quadratic generation for smooth polytopes

Let $P\subset\mathbb{R}^d$ be a lattice polytope, and let ${\mathcal A}_P=\{(\alpha,1)\in\mathbb{Z}^{d+1}:\alpha\in P\cap\mathbb{Z}^d\}$. The toric ideal of $P$ is the kernel of the map from the polynomial ring whose variables correspond to the points of ${\mathcal A}_P$ to the toric ring generated by their Laurent monomials. The polytope $P$ is smooth if, at every vertex, its primitive edge-direction vectors form a $\mathbb{Z}$-basis of the ambient lattice. Bøgvad's conjecture. The toric ideal of every smooth polytope is generated by quadratic binomials. This is a conjecture about quadratic generation of toric ideals associated with smooth polytopes, linking Gröbner-basis questions in combinatorial algebra with smooth projective toric geometry. Its resolution status is not specified in the supplied source context.