Migliore–Nagel injective conjecture for Artinian Gorenstein algebras presented by quadrics

Let $A$ be an Artinian Gorenstein algebra of socle degree at least three, presented by quadrics over a field $\mathbb{K}$ of characteristic zero. For $L\in A_1$, write $\bullet L:A_1\to A_2$ for the multiplication map. Migliore–Nagel injective conjecture. There exists $L\in A_1$ such that

$$\bullet L:A_1\to A_2$$

is injective. The conjecture concerns the first Lefschetz multiplication map for quadratic Artinian Gorenstein algebras. The paper constructs families of counterexamples to the broader weak Lefschetz conjecture, but the supplied text does not establish the status of this injective conjecture.