Uniqueness of irreducible-transformation length in Parikh rewriting

Let $\mathfrak{P}=(\Sigma,R,C)$ be a Parikh rewriting system, and suppose that $w\Rightarrow_{\mathfrak{P}}w'$ has a sequence of irreducible transformations as in Theorem~, of length $n$. Uniqueness-of-length conjecture. The length $n$ of the sequence of irreducible transformations in Theorem~ is uniquely determined by $w\Rightarrow_{\mathfrak{P}}w'$. If this conjecture is false, a counterexample should be provided. The question concerns whether different decompositions of the same Parikh rewriting transformation can have different numbers of irreducible steps; the supplied text gives no resolution.