Termination of the balanced pair algorithm under rewriting

Let $\varphi$ be a Pisot substitution on $n$ letters. Let $\tilde\varphi$ be a rewriting of $\varphi$ such that, for some letters $b,e$ in the rewritten alphabet, $\tilde\varphi(i)=b\ldots e$ for all $i$. Let $L_\lambda$ be the specified length vector for the rewritten substitution. Assume that the balanced pair algorithm for the original Pisot substitution $\varphi$ terminates. Rewriting termination conjecture. The balanced pair algorithm for the rewritten substitution $\tilde\varphi$ also terminates for the length vector $L_\lambda$. The claim concerns whether termination is preserved when a Pisot substitution is replaced by a rewriting into composite letters; the preceding example gives evidence for this expectation, but the paper states that it has not been shown.