The $q$-analog of the Markoff injectivity conjecture over the language of a balanced sequence

The Markoff injectivity conjecture states that $w\mapsto \mu (w{)}_{12}$ is injective on the set of Christoffel words where $\mu :\{\mathtt{0},\mathtt{1}{\}}^{\ast }\to {\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2\times 2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog ${\mu }_q$ of $\mu$ such that ${\mu }_q(w{)}_{12}{\mathrm{\mid }}_{q=1}=\mu (w{)}_{12}$ is the Markoff number associated to the Christoffel word $w$ when evaluated at $q=1$. We show that there exists an order ${<}_{radix}$ on $\{\mathtt{0},\mathtt{1}{\}}^{\ast }$ such that for every balanced sequence $s\in \{\mathtt{0},\mathtt{1}{\}}^{\mathbb{Z}}$ and for all factors $u,v$ in the language of $s$ with $u{<}_{radix}v$, the difference ${\mu }_q(v{)}_{12}-{\mu }_q(u{)}_{12}$ is a nonzero polynomial of indeterminate $q$ with nonnegative integer coefficients. Therefore, the map $u\mapsto {\mu }_q(u{)}_{12}$ is injective over the language of a balanced sequence. The proof uses an equivalence between balanced sequences satisfying some Markoff property and indistinguishable asymptotic pairs.