On a conjecture about the absence of an initial balanced pair for Pisot substitutions

Sellami and Sirvent conjectured that the balanced pair algorithm fails for the following pair of Pisot substitutions: $${\phi }_0:\begin{array}{l}a\mapsto abcb\mapsto ac\mapsto ac\end{array}\text{ and }{\phi }_1:\begin{array}{l}a\mapsto cbab\mapsto ac\mapsto ca\end{array}\mathrm{.}$$ The conjecture stated the balanced pair algorithm fails because there is no initial balanced pair. In the present note we prove this conjecture using a method based on simultaneous coding of the pair of the fixed points of the morphisms.