The $q$-analog of the Markoff injectivity conjecture over balanced sequences

Let $\mathcal{B}=\\{s\in\\{\mathtt{0},\mathtt{1}\\}^{\mathbb{Z}}:s\text{ is balanced}\\}$ be the set of all balanced bi-infinite binary sequences, and let

$$\mathcal{L}(\mathcal{B})=\bigcup_{s\in\mathcal{B}}\mathcal{L}(s)$$

be their language of finite factors. Let $\mu_q$ be the $q$-analog of the map assigning to a finite word $w$ the polynomial $\mu_q(w)_{12}\in\mathbb{Z}[q]$. The $q$-analog of the Markoff injectivity conjecture over balanced sequences. The map $w\mapsto\mu_q(w)_{12}$ from $\\{\mathtt{0},\mathtt{1}\\}^*$ to $\mathbb{Z}[q]$ is injective over $\mathcal{L}(\mathcal{B})$. This extends the $q$-analog from Christoffel words to the language of all balanced sequences, including periodic and Sturmian sequences; the supplied text gives no resolution.