  The $n$-cube is the poset obtained by ordering all subsets of
$\{1,\ldots,n\}$ by inclusion, and it can be partitioned into
$\binom{n}{\lfloor n/2\rfloor}$ chains, which is the minimum possible number.
Two such decompositions of the $n$-cube are called orthogonal if any two chains
of the decompositions share at most a single element. Shearer and Kleitman
conjectured in 1979 that the $n$-cube has $\lfloor n/2\rfloor+1$ pairwise
orthogonal decompositions into the minimum number of chains, and they
constructed two such decompositions. Spink recently improved this by showing
that the $n$-cube has three pairwise orthogonal chain decompositions for $n\geq
24$. In this paper, we construct four pairwise orthogonal chain decompositions
of the $n$-cube for $n\geq 60$. We also construct five pairwise edge-disjoint
chain decompositions of the $n$-cube for $n\geq 90$, where edge-disjointness is
a slightly weaker notion than orthogonality.
