Kalai's full flag conjecture for centrally symmetric polytopes

Let $P$ be a centrally symmetric $d$-dimensional polytope. A flag of $P$ is a chain of faces of $P$, one in each dimension. Kalai's full flag conjecture. $P$ has at least $2^d\cdot d!$ flags, with equality if and only if $P$ is a linear image of a Hanner polytope. The conjecture concerns the number of flags in centrally symmetric polytopes; the paper proves it for locally anti-blocking polytopes, while the general case remains open.