Markoff injectivity conjecture for Christoffel words

A Markoff triple is a positive solution of $x^2+y^2+z^2=3xyz$. A Markoff number is an element of a Markoff triple. Let $\mu$ be the monoid homomorphism from $\{\mathtt{0},\mathtt{1}\}^*$ to $\mathrm{GL}_2(\mathbb{Z})$ defined by

$$\mu(\mathtt{0})=\begin{pmatrix}2&1\\1&1\end{pmatrix},\qquad \mu(\mathtt{1})=\begin{pmatrix}5&2\\2&1\end{pmatrix}.$$

For a matrix $M$, write $M_{12}$ for its entry above the diagonal, and let $w$ range over Christoffel words. Markoff injectivity conjecture. The map

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

is injective on the set of Christoffel words. Each Markoff number is known to arise as $\mu(w)_{12}$ for some Christoffel word, so the conjecture would establish uniqueness and hence a bijection between Christoffel words and Markoff numbers. It is a longstanding question that has remained open for more than 100 years.