  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.
