Unbounded-order irreducible transformations for Parikh rewriting systems

Let $\mathfrak{P}$ be the Parikh rewriting system considered for the ternary alphabet, and let the order of an irreducible transformation denote its order in the system. An irreducible transformation is expressible as a sequence of irreducible transformations of lower order when it can be decomposed into such transformations. Unbounded-order irreducibility conjecture. Irreducible transformations of $\mathfrak{P}$ of arbitrarily large order that cannot be expressed as a sequence of irreducible transformations of lower order exist. This would show that irreducible transformations of bounded order do not suffice to describe all irreducible transformations for this system; the paper leaves the question open because its analysis only gives an exhaustive description in orders two and three.