Highly neighborly centrally symmetric spheres

In 1995, Jockusch constructed an infinite family of centrally symmetric $3$-dimensional simplicial spheres that are cs-$2$-neighborly. Here we generalize his construction and show that for all $d\geq 3$ and $n\geq d+1$, there exists a centrally symmetric $d$-dimensional simplicial sphere with $2n$ vertices that is cs-$\lceil d/2\rceil$-neighborly. This result combined with work of Adin and Stanley completely resolves the upper bound problem for centrally symmetric simplicial spheres.