Klee–Nevo–Novik–Zheng's higher symmetric stackedness conjecture

Let $P$ be a centrally symmetric simplicial $d$-polytope, and let $g_r(P)$ denote its $r$th $g$-number. Let ${\mathcal C}^*_d$ be the $d$-dimensional cross-polytope. Assume that

$$g_r(P)={d\choose r}-{d\choose {r-1}}$$

for some $3\leq r\leq \lfloor d/2\rfloor$. A polytopal complex is a collection of polytopes closed under taking faces and intersections; a cellulation of $P$ is such a complex whose union is $P$.

Klee–Nevo–Novik–Zheng's conjecture. There exists a unique polytopal complex ${\mathcal C}$ in ${\mathbb R}^d$ such that: (i) one face of ${\mathcal C}$ is the cross-polytope ${\mathcal C}^*_d$, while all other faces are simplices occurring in antipodal pairs; (ii) ${\mathcal C}$ is a cellulation of $P$; and (iii) every element of ${\mathcal C}$ of dimension at most $d-r$ is a face of $P$.

This is the proposed characterization of equality in the centrally symmetric generalized lower bound inequality for $r>2$. The $r=2$ case is known, while the higher cases are stated to be wide open.