Weak Lefschetz Property for codimension-three Gorenstein algebras

Determine whether every Artinian Gorenstein algebra A = k[x_1, x_2, x_3]/I in codimension three has the Weak Lefschetz Property, i.e., whether there exists a linear form ℓ such that ×ℓ: [A]_i → [A]_{i+1} has maximum rank in all degrees.

Background

The Weak Lefschetz Property is a central topic in the paper of graded algebras. While WLP is known in several settings (e.g., for certain complete intersections and monomial cases), its validity for all codimension-three Artinian Gorenstein algebras remains unsettled.

The authors emphasize that this question is still open, highlighting the broader landscape of unresolved Lefschetz property problems beyond the specific conic-focused results developed in the paper.