Let $P_n$ denote the poset with elements $P_n=\{1,...,n\}$ ordered by divisibility and let $L_n$ denote the distributive lattice of order ideals of $P_n$, whose elements should correspond to primitive subsequences of $\{1,...,n\}$, see https://oeis.org/A051026.

Question: Is $L_n$ Sperner? What is the width of $L_n$?

The width sequence start for $n \geq 2$ with 1,2,2,4,4,8,10,15,21,40,45,87 and the poset is indeed Sperner for $n \leq 15$.