Uniqueness conjecture for generalized Markov values

Let $\mathcal M_{0,0,0}$ denote the classical discrete Markov spectrum, and let $\mathcal L(\alpha)$ denote the Markov-Lagrange value associated with an irrational $\alpha$. Two irrationals are $GL(2,\mathbb Z)$-equivalent when there exists a matrix in $GL(2,\mathbb Z)$ relating them by a fractional linear transformation. The generalized uniqueness conjecture. For any element $L\in \mathcal M_{0,0,0}$, if $L=\mathcal L(\alpha)=\mathcal L(\beta)$, $\alpha$ and $\beta$ are $GL(2,\mathbb Z)$-equivalent. That is, there exists

$$\begin{bmatrix} a&b\\c&d \end{bmatrix}\in GL(2,\mathbb Z)$$

such that

$$\alpha=\frac{a\beta+b}{c\beta+d}.$$

This is stated as an equivalent formulation of Frobenius's uniqueness conjecture. The supplied text gives no resolution status for this formulation.