The $q$-analog of the Markoff injectivity conjecture

Let $\mu_q$ be the $q$-analog of the monoid homomorphism used to define $q$-Markoff numbers, and let $\mu_q(w)_{12}\in\mathbb{Z}[q]$ denote the corresponding entry above the diagonal. Let $\\{\mathtt{0},\mathtt{1}\\}^*$ be the set of finite binary words, and let Christoffel words be the usual Christoffel words over this alphabet. $q$-analog of the Markoff injectivity conjecture. The map

$$w\longmapsto\mu_q(w)_{12}$$

from $\\{\mathtt{0},\mathtt{1}\\}^*$ to $\mathbb{Z}[q]$ is injective over the set of Christoffel words. This is a $q$-analogue of the classical Markoff injectivity conjecture; the text notes that it is weaker than the classical conjecture because distinct polynomials can agree at $q=1$, and no resolution is supplied.