Weak Lefschetz conjecture for Gorenstein algebras presented by quadrics

Let $R/I$ be an artinian Gorenstein algebra presented by quadrics, with socle degree at least $3$, and suppose that $R$ is defined over a field of characteristic zero. The WLP conjecture. Then $R/I$ has the Weak Lefschetz Property: multiplication by a general linear form has maximal rank in every degree. The conjecture is posed as a stronger characteristic-zero form of the injectivity conjecture; it is known for complete intersections of quadrics and for complete intersections of at most four quadrics, but remains open in general.