Existence of a graphic-matroid counterexample failing at degree 2

Determine whether there exists a connected graph whose associated algebra defined by its basis generating polynomial fails the strong Lefschetz property at degree 2.

Background

The paper studies the Maeno–Numata conjecture, which asserts that the Artinian Gorenstein algebra associated with the basis generating polynomial of any matroid has the strong Lefschetz property. The paper disproves the conjecture for graphic matroids by exhibiting connected graphs whose associated algebras fail the strong Lefschetz property at degree 3.

The authors explicitly leave unresolved whether an analogous graphic-matroid counterexample exists at degree 2. In the paper’s notation, failure at degree 2 means that the multiplication map from the degree-2 component to its Poincaré-dual component cannot be an isomorphism for any linear Lefschetz element. The later discussion reports that every graph with eight or fewer vertices has the strong Lefschetz property at degree 2, but does not resolve the question for arbitrary connected graphs.