The conjecture that join–meet ideals of nonmodular lattices are not linearly related

Let $L$ be a nonmodular lattice, and let $I_L$ denote its join–meet ideal. An ideal is linearly related when its linear syzygies generate all of its syzygies. Nonmodular-lattice join–meet ideal conjecture. The ideal $I_L$ is not linearly related.

The preceding example verifies this for each nonmodular lattice with at most six elements. The conjecture asks whether the same obstruction holds for every nonmodular lattice.