The colored strong Lefschetz property conjecture for balanced simplicial spheres

A simplicial complex is balanced if its vertices admit a proper coloring by $d$ colors, where every facet has exactly one vertex of each color; its Stanley–Reisner ring is denoted by ${\mathbb F}[\Delta]$. The colored strong Lefschetz property (colored SLP) means that an Artinian reduction of ${\mathbb F}[\Delta]$ by a colored system of parameters has the strong Lefschetz property.

Colored SLP conjecture. Any balanced simplicial sphere, or at least any balanced simplicial polytope, has the colored SLP over a field of characteristic $0$.

This conjecture proposes a higher-dimensional analogue of the preceding result for $(2,1)$-balanced simplicial $2$-spheres, despite the failure of the corresponding statement for general ${\mathbf a}$-balanced simplicial polytopes. The source does not provide a resolution.