Existence of Lefschetz conics for non-general height-3 Gorenstein algebras

Determine whether every (not necessarily general) Artinian Gorenstein algebra A = k[x_1, x_2, x_3]/I admits at least one degree-2 form C such that the multiplication maps ×C: [A]_i → [A]_{i+2} have maximum rank in every degree, i.e., whether A possesses a Lefschetz conic.

Background

The authors show that for general height-3 Gorenstein algebras with appropriate conditions on the Hilbert function (e.g., decreasing-type g-vectors), the non-Lefschetz locus of conics has the expected codimension, and thus there exist Lefschetz conics. However, without the generality assumption, this existence is not guaranteed by the methods in the paper.

They explicitly note that for arbitrary (non-general) Artinian Gorenstein algebras in three variables, it is unknown whether any Lefschetz conic exists; if none exists, the non-Lefschetz locus could be all of P^5 (codimension zero).