  In 1993 Stanley showed that if a simplicial complex is acyclic over some
field, then its face poset can be decomposed into disjoint rank $1$ boolean
intervals whose minimal faces together form a subcomplex. Stanley further
conjectured that complexes with a higher notion of acyclicity could be
decomposed in a similar way using boolean intervals of higher rank. We provide
an explicit counterexample to this conjecture. We also prove a version of the
conjecture for boolean trees and show that the original conjecture holds when
this notion of acyclicity is as high as possible.
