Babson–Billera–Chan cubical generalized lower bound conjecture

Let $d$ be a positive integer. For a cubical $d$-polytope $Q$, let $g^c(Q)$ be its cubical $g$-vector, and let $\mathcal{C}_d$ be the minimal closed cone containing all such vectors. Let $\mathcal{A}_d$ denote the nonnegative orthant in $\mathbb{R}^{\lfloor d/2\rfloor}$. Babson–Billera–Chan's cubical generalized lower bound conjecture.

$$\mathcal{A}_d=\mathcal{C}_d.$$

The paper proves the inclusion $\mathcal{A}_d\subseteq\mathcal{C}_d$, so the conjecture is equivalent to the reverse inclusion. It concerns the possible $f$-vectors of cubical polytopes and, in particular, would imply nonnegativity of every coordinate of the cubical $g$-vector.