h-vector criterion for purity of zero-sumfree complexes

Let $\Delta_{n,\ell}$ be the simplicial complex of $\ell$-zero-sumfree subsets, and let $h_1,\ldots,h_d$ be the relevant entries of its $h$-vector. A simplicial complex is pure when all of its facets have the same dimension. h-vector purity conjecture. The complex $\Delta_{n,\ell}$ is pure if $h_i\geq0$ for every $i=1,\ldots,d$. The conjecture proposes that nonnegativity of these $h$-vector entries is sufficient for purity; the source gives no resolution status.