Construct an explicit non-polytopal Bier sphere

Construct a specific simplicial complex K on a finite vertex set [m], with K ≠ Δ_[m], such that the Bier sphere Bier(K) is non-polytopal; that is, Bier(K) is not combinatorially equivalent to the boundary complex of any convex polytope.

Background

Bier spheres are defined as deleted joins of a simplicial complex K with its Alexander dual K^∨ and are known to be PL-spheres. They admit canonical starshaped realizations and play a significant role in toric topology, connecting combinatorial data of simplicial complexes with toric and moment-angle manifolds.

It has been observed (following Matoušek) that asymptotically almost all Bier spheres are non-polytopal as the number of vertices grows. Despite this probabilistic prevalence, a concrete example of a non-polytopal Bier sphere has not been identified, leaving an explicit construction problem open.