Order-two decomposition conjecture for the ternary Parikh rewriting system

Let $\mathfrak{P}=(\Sigma,R,\{abc\})$, where

$$R=\{ac\rightarrow ca\}\cup\{abwba\rightarrow bawab\mid w\in\Sigma^*\}\cup\{bcwcb\rightarrow cbwbc\mid w\in\Sigma^*\}.$$

For words $w,w'\in\Sigma^*$, let $d(w,w')$ denote their distance, and let an irreducible transformation of order at most two mean an irreducible transformation whose order is at most $2$. Order-two decomposition conjecture. If $w\Rightarrow_{\mathfrak{P}}w'$ and $w$ and $w'$ differ in $2*d(w,w')$ positions, then there is a sequence of irreducible transformations of order at most $2$ that takes $w$ into $w'$. The claim proposes a bounded-order decomposition for transformations satisfying the stated positional-difference condition; the source appends “Why should one be interested in this?” and provides no evidence of resolution, so the conjecture remains open.