Centrally symmetric Cohen--Macaulay complexes and a conjecture of Stanley

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 $1\leq i\leq d-1$, then $h_j(\Delta)=\binom{d}{j}$ for all $j\geq i$. This note proves Stanley's conjecture.