  In 1987, Stanley conjectured that if a centrally symmetric Cohen--Macaulay
simplicial complex $\Delta$ of dimension $d-1$ satisfies
$h_i(\Delta)=\binom{d}{i}$ for some $i\geq 1$, then $h_j(\Delta)=\binom{d}{j}$
for all $j\geq i$. Much more recently, Klee, Nevo, Novik, and Zheng conjectured
that if a centrally symmetric simplicial polytope $P$ of dimension $d$
satisfies $g_i(\partial P)=\binom{d}{i}-\binom{d}{i-1}$ for some $d/2\geq i\geq
1$, then $g_j(\partial P)=\binom{d}{j}-\binom{d}{j-1}$ for all $d/2\geq j\geq
i$. This note uses stress spaces to prove both of these conjectures.
