The polytopal cellulation conjecture for centrally symmetric polytopes with minimal higher g-number

Let $P$ be a centrally symmetric simplicial $d$-polytope, and suppose that for some $3\leq r\leq\left\lfloor\frac d2\right\rfloor$,

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

The polytopal cellulation 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 $\mathcal C^*_d$, and all other faces are simplices occurring in antipodal pairs; (ii) $\mathcal C$ cellulates $P$, meaning

$$\bigcup_{C\in\mathcal C}C=P;$$

(iii) every element of $\mathcal C$ of dimension at most $d-r$ is a face of $P$. Moreover, the simplices of $\mathcal C$ are precisely all proper faces of $P$ together with all simplices $\operatorname{conv}(U)$, where $U\subset V(P)$ and the $(d-r)$-skeleton of $\overline U$ is contained in $\partial P$. This generalizes the equality case suggested by the generalized lower bound theorem and would imply the higher cs lower bound conjecture.