Calling this poset $M(n)$, the fact that it has the Sperner property was conjectured in B. Lindström, "A conjecture on a theorem similar to Sperner's", Combinatorial Structures and Their Applications, p. 241.

It turns out that $M(n)$ has the $k$-Sperner property for all $k$, see R. Stanley's paper "Weyl groups, the hard Lefschetz theorem, and the Sperner property", in the section on the type $B_n$ the properties of $M(n)$ are shown to follow from the general theorems about posets derived from complex semisimple algebraic groups by quotienting by a parabolic subgroup and using the Bruhat order. See also Stanley's article "Some applications of algebra to combinatorics". The papers are available at his website.

Calling this poset $M(n)$, the fact that it has the Sperner property was conjectured in B. Lindström, "A conjecture on a theorem similar to Sperner's", Combinatorial Structures and Their Applications, p. 241.

It turns out that $M(n)$ has the $k$-Sperner property for all $k$, see R. Stanley's paper "Weyl groups, the hard Lefschetz theorem, and the Sperner property", in the section on the type $B_n$ the properties of $M(n)$ are shown to follow from the general theorems about algebraic posets, the main ingredient being the hard Lefschetz theorem. See also Stanley's article "Some applications of algebra to combinatorics". The papers are available at his website.