Resolving Stanley's conjecture on $k$-fold acyclic complexes

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 subcomplex. further conjectured complexes with higher notion of acyclicity could in similar way using rank. We provide an explicit counterexample to this conjecture. also prove version the conjecture for trees and show original holds when as high possible.Mathematics Subject Classifications: 05E45, 55U10