Let $G(\mathcal{P}(n),E)$ be the undirected graph for the power set of $[n]$ elements under the inclusion relation (i.e. a poset). The width of this poset - which is defined as the size of the maximum antichain- is $k_1=\binom{n}{n/2}$( per Sperner's theorem when $n$ is even).
Let $H(\mathcal{P}(n),\hat{E})$ be a bipolar orientation of $G$. $H$ defines a poset with width $k_2$. Is it always the case that $k_1\geq k_2$? Because from trail and error and experimenting on $[3]$ it seems that's the case. Any hints would be appreciated.