Colored SLP conjecture for balanced spheres

Prove that for any balanced d-dimensional simplicial sphere Δ and its colored system of parameters θ1, …, θd+1 (defined by summing variables in each color class), the quotient algebra R/(I_Δ + (θ1, …, θd+1)) satisfies the strong Lefschetz property.

Background

For a balanced d-dimensional complex Δ with a (d+1)-coloring ρ of its 1-skeleton, Stanley showed that the linear forms θi obtained by summing variables of color i form a system of parameters (the colored sop).

The Lefschetz properties of the artinian reduction A = R/(I_Δ + (θ1, …, θd+1)) have been investigated; in particular, a conjecture asserts that A should satisfy the strong Lefschetz property (SLP), often referred to as the colored SLP.

This paper relates colored sops to unexpected systems of parameters and balancedness, reinforcing the interest in resolving the colored SLP.