The rational non-freeness conjecture for two parabolic generators

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 $\alpha$ free when these generators freely generate $G_{\alpha}$. Rational non-freeness conjecture. Every rational number $\alpha$ with $|\alpha|<4$ is non-free. This is a long-standing open problem concerning the characterization of non-free parameters; although non-free numbers are known to be dense in the relevant real interval, determining whether a given rational number is free remains difficult.