Cobham-type conjecture for independent substitutions

Let $\sigma$ and $\tau$ be two independent substitutions, each having a proper fixed point that is mapped by a letter-to-letter morphism onto a sequence $x$. A sequence is ultimately periodic if it agrees with a periodic sequence from some position onward.

Independent-substitution conjecture. Then $x$ is ultimately periodic.

This conjecture is a substitution-theoretic analogue of Cobham's theorem. The paper addresses it with partial answers, including results for substitutions satisfying additional growth and bounded-gap hypotheses; the general assertion remains open in the supplied context.