Migliore–Nagel weak Lefschetz conjecture for Artinian Gorenstein algebras

Let $A=\bigoplus_i A_i$ be an Artinian Gorenstein algebra presented by quadrics over a field $\mathbb{K}$ of characteristic zero. The Weak Lefschetz Property means that there exists $L\in A_1$ such that every multiplication map

$$\bullet L:A_i\to A_{i+1}$$

has maximal rank. Migliore–Nagel weak Lefschetz conjecture. Every such algebra has the Weak Lefschetz Property. The abstract states that the paper provides families of counterexamples to this conjecture, so it is refuted.