Existence of unexpected systems of parameters for collapsible complexes

Determine whether there exists any collapsible simplicial complex that admits an unexpected system of parameters in the sense of Definition 3.1 (for some choice of a and total degree).

Background

The paper proves a strong nonexistence result for collapsible complexes: if Δ is d-dimensional Cohen–Macaulay and collapsible, then for fixed a > 0 there is no a-unexpected system of parameters of total degree t > d(a−1).

Beyond this bound, however, the authors remark that it is unknown whether any collapsible complex has an unexpected sop at all, highlighting a gap left open by the nonexistence theorem.