Cohen–Macaulayness of coordinate rings for all collections of cells

Ascertain whether the coordinate ring K[P]=S_P/I_P is Cohen–Macaulay for every collection of cells P, or equivalently, determine whether there exists any collection of cells whose coordinate ring fails to be Cohen–Macaulay.

Background

After proving Cohen–Macaulayness for several non-prime classes (zig-zag collections and closed paths with zig-zag walks), the authors note that no counterexample is currently known.

This motivates a general question about the universality of the Cohen–Macaulay property across all collections of cells, which, if resolved affirmatively, would unify many scattered results in the literature.