Following Ville Salo's comments I came across upon Pansiot's paper, DECIDABILITY OF PERIODICITY FOR INFINITE WORDS. The papers shows an algorithm to decide whether an infinite word fixed by a substitution is aperiodic relying heavily on the Coven-Hedlund theorem regarding complexity of an infinite word.
If I am not mistaken, the crux of the paper is Lemma 2 discussing whether finite subwords have unique extension to a larger words. If I am not mistaken, in the case I am interested the lemma simplifies to the following:
Let $S:A\to A^+$ be an injective substition such that $\vert S^n(a)\vert\to \infty$ for all $a\in A$. Then $\omega=\lim_{n\to
> \infty} S^n(a)$ is aperiodic if and only if $c_2(\omega)>c_1(\omega)$,
where $c_k(\omega)$ is the number of $k$-subwords in $\omega$.
The original statement talks about having unique-extension\prolongablity of sub-words of the form $au\in A^+$, where $a\in A$ satisfies $\vert S^n(a)\vert\to \infty$ and $u\in A^*$. Since I assume every letter is growing, I can take the lemma with $u$ being the empty word.
I post this as an answer since I don't think I should add this in an edit to the original question. I would welcome and appreciate any corrections or pointing out if this argument is incorrect.
