The extension conjecture for the Winder existential ordering

Let $[n]=\{1,\ldots,n\}$ and let $\Delta\subseteq2^{[n]}$ be a strongly acyclic 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 $\prec_W$ by $A\prec_W B$ if and only if it is not the case that $B\leq_W A$. An extension of $\prec_W$ is a total ordering on $2^{[n]}$ containing every relation in $\prec_W$; a qualitative probability order is a total order satisfying de Finetti's axiom. Winder extension conjecture. If $\Delta$ is strongly acyclic then there exists an extension of $\prec_W$ to a qualitative probability order. This is presented as a slightly stronger version of the preceding characterization conjecture. The preceding lemmas provide support by showing that $\prec_W$ respects membership in $\Delta$ and satisfies the required cancellation property, but the existence of the extension remains open in the supplied text.