The ball-or-sphere conjecture for polarizations of Artinian monomial ideals

Let $I$ be an Artinian monomial ideal and let $J$ be a polarization of $I$. Write $4\Delta(J)4$ for the simplicial complex associated to $J$. Ball-or-sphere conjecture. The simplicial complex $\Delta(J)$ is a simplicial ball, except when $I$ is a complete intersection, in which case it is a simplicial sphere.

The conjecture would follow from constructibility of the simplicial complexes associated to polarizations. It is known for standard polarizations and for several classes including letterplace ideals, and for polarizations in three variables or for squares of graded maximal ideals through shellability results.