The equality characterization for the boundary $h_2$-bound

Let $\Delta$ be a connected $(d-1)$-dimensional homology manifold with nonempty orientable boundary, where $d\geq 4$. For each face $F$ of $\Delta$, let $\operatorname{lk}_\Delta(F)$ denote its link. A simplicial complex is stacked if it is obtained from a simplex or a simplicial sphere by successive facet subdivisions; thus the relevant links are stacked polytopes or stacked spheres. Equality characterization. Equality occurs in the paper's boundary $h_2$ bound if and only if every link of $\Delta$ is combinatorially equivalent to a stacked polytope or a stacked sphere. This would characterize all equality cases in the stated boundary inequality; the preceding constructions provide examples with stacked-polytope boundary links and stacked-sphere interior links, but the general equivalence is not established in the supplied text.