  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.
