The strong acyclicity characterization of initial segment complexes

Let $[n]=\{1,\ldots,n\}$ and let $\Delta\subseteq 2^{[n]}$ be a simplicial complex. Define the Winder desirability relation $\leq_W$ on $2^{[n]}$ by $A\leq_W B$ if and only if, for every $Z\subseteq[n]\setminus((A\setminus B)\cup(B\setminus A))$,

$$(A\setminus B)\cup Z\notin\Delta\Rightarrow(B\setminus A)\cup Z\notin\Delta.$$

Define the Winder existential ordering by $A\prec_W B$ if and only if it is not the case that $B\leq_W A$. The complex $\Delta$ is strongly acyclic when there is no cycle $A_1\prec_W A_2\prec_W\cdots\prec_W A_k\prec_W A_1$ for any $k$. An initial segment complex is one of the form $\Delta(\preceq,T)$ for a qualitative probability order $\preceq$ on $2^{[n]}$ and some $T\in2^{[n]}$. Strong acyclicity conjecture. A simplicial complex $\Delta$ is an initial segment complex if and only if it is strongly acyclic. Theorem 3.1 establishes the necessary direction: every initial segment complex is strongly acyclic. The converse is supported by subsequent lemmas showing that the Winder existential ordering has the cancellation property needed for extension to a qualitative probability order, but it is not established here.