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 $\mu:\{\mathtt{0},\mathtt{1}\}^*\to\mathrm{SL}_2(\mathbb{Z})$ is a certain homomorphism and $M_{12}$ is the entry above the diagonal of a $2\times2$ matrix $M$. Recently, Leclere and Morier-Genoud (2021) proposed a $q$-analog $\mu_q$ of $\mu$ such that $\mu_{q}(w)_{12}|_{q=1}=\mu(w)_{12}$ is the Markoff number associated to the Christoffel word $w$ when evaluated at $q=1$. We show that there exists an order $<_{radix}$ on $\{\mathtt{0},\mathtt{1}\}^*$ 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.