The dense rational non-freeness conjecture

Let $G_{\alpha}=\left\langle \begin{bmatrix}1&1\\0&1\end{bmatrix},\begin{bmatrix}1&0\alpha&1\end{bmatrix}\right\rangle<\operatorname{SL}_2(\mathbb{C})$, and call a parameter non-free when these generators do not freely generate the group. Dense rational non-freeness conjecture. There exists a dense subset $S\subset\mathbb{Q}\cap(-4,4)$ consisting of non-free numbers. The paper notes that the rational non-freeness conjecture remains open and that the authors' results give only limited information about the distribution of non-free rational numbers.