Converse of the balancedness criterion for regular staged trees

Let a staged tree be regular when, for every pair of vertices in the same stage and every pair of outgoing edges, the corresponding subtree monomials satisfy the required equality of cross-products. Let it be balanced when these cross-product identities hold for every such pair of vertices and edges. Converse of the balancedness criterion. All regular staged trees are balanced. The paper states that this converse holds for all binary trees with three levels; whether it holds for regular staged trees in the stated generality is left as an open question.