SLP for algebras defined by unexpected systems of parameters (algebraic g-conjecture direction)

Determine whether, for every d-dimensional Q-homology sphere Δ on vertex set [n] and any a-unexpected system of parameters θ1, …, θd+1 of Δ, the algebra C[x1, …, xn]/(I_Δ + (θ1, …, θd+1)) satisfies the strong Lefschetz property.

Background

The paper introduces unexpected systems of parameters (sops) for simplicial complexes, linking them to Lefschetz properties and to combinatorial structures such as balanced complexes.

Two major families of unexpected sops are known (universal sops from elementary symmetric polynomials and colored sops for balanced homology spheres). Motivated by these and computational evidence, the authors ask whether any a-unexpected sop for a Q-homology sphere yields an algebra with SLP, a question paralleling themes of the algebraic g-conjecture.