Local dimension of a Boolean lattice

For every integer $n$ with $n\ge 4$, we prove that the local dimension of a poset consisting of all the subsets of $\{1,\dots ,n\}$ equipped with the inclusion relation is strictly less than $n$, answering a question of Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang (Eur. J. Comb. 2020). We also study several related problems.