Generalized lower bound conjecture for complexes with bounded missing-face dimension

Let $d ot obreak\geq 0$ and $0<i\leq d+1$ be integers. Let $\mathcal{C}(i,d)$ be the family of $d$-dimensional simplicial complexes with nonzero reduced $d$-homology and no missing faces of dimension greater than $i$. Write $d+1=qi+r$ with $q\geq 0$ and $1\leq r\leq i$, and let $S(i,d)$ be the join of $q$ copies of the boundary of the $i$-simplex and the boundary of the $r$-simplex. Let $\operatorname{f}_j(\Delta)$ denote the number of $j$-dimensional faces of $\Delta$. Generalized lower bound conjecture. For $\Delta\in\mathcal{C}(i,d)$,

$$\operatorname{f}_j(\Delta)\geq \operatorname{f}_j(S(i,d))$$

for every $j$. Moreover, if equality is attained for every $j$, then $\Delta=S(i,d)$. The source presents this as a conjectural strengthening of known extremal results; the condition $i\mid(d+1)$ is noted as an artifact of the proof in the surrounding theorem, so the full assertion remains open.