Probability-function polytopes of disjoint unions

Determine whether, for finite posets P_1 and P_2 and their disjoint union P=P_1+P_2, the probability-functions poset A_P and the probability-functions polytope Q_P can be described in terms of A_{P_1}, A_{P_2}, Q_{P_1}, and Q_{P_2}.

Background

The paper establishes product decompositions for ordinal sums: the antichain poset of P\oplus Q is isomorphic to the disjoint union of the antichain posets of P and Q, and the corresponding probability-functions polytope is combinatorially equivalent to the product of the two component polytopes. The authors ask whether an analogous structural description exists for disjoint unions.

The immediately following example shows that a description based only on the separate components cannot be straightforward. For P_1=P_2=C_2, each component has a trivial probability-functions polytope and an empty antichain poset, whereas the disjoint union P_1+P_2 has a nontrivial antichain poset and a four-dimensional probability-functions polytope. Thus, any satisfactory description must account for interactions between the components created by incomparability across the disjoint union.