Sparla's Euler-characteristic conjecture for centrally symmetric manifolds

Let $M$ be a centrally symmetric combinatorial $2r$-dimensional manifold with $2k$ vertices, and let $\chi(M)$ denote its Euler characteristic. Let the $r$-skeleton of the $k$-dimensional cross polytope mean the subcomplex consisting of all faces of dimension at most $r$.

Sparla's conjecture. The inequality

$$(-1)^r\binom{2r+1}{r+1}(\chi(M)-2)\leq 4^{r+1}\binom{\frac{1}{2}(k-1)}{r+1}$$

holds. Moreover, equality is attained if and only if $M$ contains the $r$-skeleton of the $k$-dimensional cross polytope.

Both assertions were proved under the additional restriction that $M$ has at least $6r+4$ vertices. Thus the conjecture is solved in that restricted range, while the unrestricted assertion stated above is not established by the supplied evidence.